Digital options having demand-based, adjustable returns, and trading exchange therefor

ABSTRACT

Methods and systems for conducting demand-based trading are described. In one embodiment, states are established, each state corresponding to at least one possible outcome of an event of economic significance. An investment amount may be determined as a function of a selected outcome, a desired payout, and a total amount invested in the states. In another embodiment, an investment amount may be determined as a function of parameters of a financial product. In another embodiment, a payout may be determined as a function of an investment amount, a selected outcome, a total amount invested in the states, and an identification of a state corresponding to an observed outcome of the event.

RELATED APPLICATIONS

This application is a divisional application of U.S. patent applicationSer. No. 09/950,498, filed Sep. 10, 2001, which is acontinuation-in-part of U.S. patent application Ser. No. 09/809,025,filed Mar. 16, 2001, which issued as U.S. Pat. No. 7,225,153; which is acontinuation-in-part of U.S. patent application Ser. No. 09/744,816,which was attributed a filing date of Apr. 3, 2001 and which issued asU.S. Pat. No. 7,389,262; which was the United States national stageapplication under 35 U.S.C. §371 of Patent Cooperation Treatyapplication serial number PCT/US00/19447, filed Jul. 18, 2000; which isa continuation-in-part of U.S. patent application Ser. No. 09/448,822,filed Nov. 24, 1999, which issued as U.S. Pat. No. 6,321,212; whichclaims the benefit, under 35 U.S.C. §119(e), of U.S. provisional patentapplication Ser. No. 60/144,890, filed Jul. 21, 1999. Each of theapplications referred to in this paragraph is incorporated by referencein its entirety into this application.

COPYRIGHT NOTICE

This document contains material that is subject to copyright protection.The applicant has no objection to the facsimile reproduction of thispatent document, as it appears in the U.S. Patent and Trademark Office(PTO) patent file or records or in any publication by the PTO orcounterpart foreign or international instrumentalities. The applicantotherwise reserves all copyright rights whatsoever.

FIELD OF THE INVENTION

This invention relates to systems and methods for demand-based trading.More specifically, this invention relates to methods and systems fortrading financial products, including digital options, havingdemand-based adjustable returns, and systems and methods for determiningthose returns.

BACKGROUND OF THE INVENTION

With the rapid increase in usage and popularity of the public Internet,the growth of electronic Internet-based trading of securities has beendramatic. In the first part of 1999, online trading via the Internet wasestimated to make up approximately 15% of all stock trades. This volumehas been growing at an annual rate of approximately 50%. High growthrates are projected to continue for the next few years, as increasingvolumes of Internet users use online trading accounts.

Online trading firms such as E-Trade Group, Charles Schwab, andAmeritrade have all experienced significant growth in revenues due toincreases in online trading activity. These companies currently offerInternet-based stock trading services, which provide greater convenienceand lower commission rates for many retail investors, compared totraditional securities brokerage services. Many expect online trading toexpand to financial products other than equities, such as bonds, foreignexchange, and financial instrument derivatives.

Financial products such as stocks, bonds, foreign exchange contracts,exchange traded futures and options, as well as contractual assets orliabilities such as reinsurance contracts or interest-rate swaps, allinvolve some measure of risk. The risks inherent in such products are afunction of many factors, including the uncertainty of events, such asthe Federal Reserve's determination to increase the discount rate, asudden increase in commodity prices, the change in value of anunderlying index such as the Dow Jones Industrial Average, or an overallincrease in investor risk aversion. In order to better analyze thenature of such risks, financial economists often treat the real-worldfinancial products as if they were combinations of simpler, hypotheticalfinancial products. These hypothetical financial products typically aredesigned to pay one unit of currency, say one dollar, to the trader orinvestor if a particular outcome among a set of possible outcomesoccurs. Possible outcomes may be said to fall within “states,” which aretypically constructed from a distribution of possible outcomes (e.g.,the magnitude of the change in the Federal Reserve discount rate) owingto some real-world event (e.g., a decision of the Federal Reserveregarding the discount rate). In such hypothetical financial products, aset of states is typically chosen so that the states are mutuallyexclusive and the set collectively covers or exhausts all possibleoutcomes for the event. This arrangement entails that, by design,exactly one state always occurs based on the event outcome.

These hypothetical financial products (also known as Arrow-Debreusecurities, state securities, or pure securities) are designed toisolate and break-down complex risks into distinct sources, namely, therisk that a distinct state will occur. Such hypothetical financialproducts are useful since the returns from more complicated securities,including real-world financial products, can be modeled as a linearcombination of the returns of the hypothetical financial products. See,e.g., R. Merton, Continuous-Time Finance (1990), pp. 441 ff. Thus, suchhypothetical financial products are frequently used today to provide thefundamental building blocks for analyzing more complex financialproducts.

In recent years, the growth in derivatives trading has also beenenormous. According to the Federal Reserve, the annualized growth ratein foreign exchange and interest rate derivatives turnover alone isrunning at about 20%. Corporations, financial institutions, farmers, andeven national governments and agencies are all active in the derivativesmarkets, typically to better manage asset and liability portfolios,hedge financial market risk, and minimize costs of capital funding.Money managers also frequently use derivatives to hedge and undertakeeconomic exposure where there are inherent risks, such as risks offluctuation in interest rates, foreign exchange rates, convertibilityinto other securities or outstanding purchase offers for cash orexchange offers for cash or securities.

Derivatives are traded on exchanges, such as the option and futurescontracts traded on the Chicago Board of Trade (“CBOT”), as well asoff-exchange or over-the-counter (“OTC”) between two or more derivativecounterparties. On the major exchanges that operate trading activity inderivatives, orders are typically either transmitted electronically orvia open outcry in pits to member brokers who then execute the orders.These member brokers then usually balance or hedge their own portfolioof derivatives to suit their own risk and return criteria. Hedging iscustomarily accomplished by trading in the derivatives' underlyingsecurities or contracts (e.g., a futures contract in the case of anoption on that future) or in similar derivatives (e.g., futures expiringin different calendar months). For OTC derivatives, brokers or dealerscustomarily seek to balance their active portfolios of derivatives inaccordance with the trader's risk management guidelines andprofitability criteria.

Broadly speaking then, there are two widely utilized means by whichderivatives are currently traded: (1) order-matching and (2) principalmarket making. Order matching is a model followed by exchanges such asthe CBOT or the Chicago Mercantile Exchange and some newer onlineexchanges. In order matching, the exchange coordinates the activities ofbuyers and sellers so that “bids” to buy (i.e., demand) can be pairedoff with “offers” to sell (i.e., supply). Orders may be matched bothelectronically and through the primary market making activities of theexchange members. Typically, the exchange itself takes no market riskand covers its own cost of operation by selling memberships to brokers.Member brokers may take principal positions, which are often hedgedacross their portfolios.

In principal market making, a bank or brokerage firm, for example,establishes a derivatives trading operation, capitalizes it, and makes amarket by maintaining a portfolio of derivatives and underlyingpositions. The market maker usually hedges the portfolio on a dynamicbasis by continually changing the composition of the portfolio as marketconditions change. In general, the market maker strives to cover itscost of operation by collecting a bid-offer spread and through the scaleeconomies obtained by simultaneously hedging a portfolio of positions.As the market maker takes significant market risk, its counterpartiesare exposed to the risk that it may go bankrupt. Additionally, while intheory the principal market making activity could be done over a widearea network, in practice derivatives trading is today usuallyaccomplished via the telephone. Often, trades are processed laboriously,with many manual steps required from the front office transaction to theback office processing and clearing.

In theory—that is, ignoring very real transaction costs (describedbelow)—derivatives trading is, in the language of game theory, a “zerosum” game. One counterparty's gain on a transaction should be exactlyoffset by the corresponding counterparty's loss, assuming there are notransaction costs. In fact, it is the zero sum nature of the derivativesmarket which first allowed the well-known Black-Scholes pricing model tobe formulated by noting that a derivative such as an option could bepaired with an exactly offsetting position in the underlying security soas to eliminate market risk over short periods of time. It is this “noarbitrage” feature that allows market participants using sophisticatedvaluation models to mitigate market risk by continually adjusting theirportfolios. Stock markets, by contrast, do not have this zero sumfeature, as the total stock or value of the market fluctuates due tofactors such as interest rates and expected corporate earnings, whichare “external” to the market in the sense that they cannot readily behedged.

The return to a trader of a traditional derivative product is, in mostcases, largely determined by the value of the underlying security,asset, liability or claim on which the derivative is based. For example,the value of a call option on a stock, which gives the holder the rightto buy the stock at some future date at a fixed strike price, variesdirectly with the price of the underlying stock. In the case ofnon-financial derivatives such as reinsurance contracts, the value ofthe reinsurance contract is affected by the loss experience on theunderlying portfolio of insured claims. The prices of traditionalderivative products are usually determined by supply and demand for thederivative based on the value of the underlying security (which isitself usually determined by supply and demand, or, as in the case ofinsurance, by events insured by the insurance or reinsurance contract).

At present, market-makers can offer derivatives products to theircustomers in markets where:

-   -   Sufficient natural supply and demand exist    -   Risks are measurable and manageable    -   Sufficient capital has been allocated        A failure to satisfy one or more of these conditions in certain        capital markets may inhibit new product development, resulting        in unsatisfied customer demand.

Currently, the costs of trading derivative securities (both on and offthe exchanges) and transferring insurance risk are considered to be highfor a number of reasons, including:

-   -   (1) Credit Risk: A counterparty to a derivatives (or insurance        contract) transaction typically assumes the risk that its        counterparty will go bankrupt during the life of the derivatives        (or insurance) contract. Margin requirements, credit monitoring,        and other contractual devices, which may be costly, are        customarily employed to manage derivatives and insurance        counterparty credit risk.    -   (2) Regulatory Requirements: Regulatory bodies, such as the        Federal Reserve, Comptroller of the Currency, the Commodities        Futures Trading Commission, and international bodies that        promulgate regulations affecting global money center banks        (e.g., Basle Committee guidelines) generally require        institutions dealing in derivatives to meet capital requirements        and maintain risk management systems. These requirements are        considered by many to increase the cost of capital and barriers        to entry for some entrants into the derivatives trading        business, and thus to increase the cost of derivatives        transactions for both dealers and end users. In the United        States, state insurance regulations also impose requirements on        the operations of insurers, especially in the property-casualty        lines where capital demands may be increased by the requirement        that insurers reserve for future losses without regard to        interest rate discount factors.    -   (3) Liquidity: Derivatives traders typically hedge their        exposures throughout the life of the derivatives contract.        Effective hedging usually requires that an active or liquid        market exist, throughout the life of the derivative contract,        for both the underlying security and the derivative. Frequently,        especially in periods of financial market shocks and        disequilibria, liquid markets do not exist to support a        well-functioning derivatives market.    -   (4) Transaction Costs: Dynamic hedging of derivatives often        requires continual transactions in the market over the life of        the derivative in order to reduce, eliminate, and manage risk        for a derivative or portfolio of derivative securities. This        usually means paying bid-offers spreads for each hedging        transaction, which can add significantly to the price of the        derivative security at inception compared to its theoretical        price in absence of the need to pay for such spreads and similar        transaction costs.    -   (5) Settlement and Clearing Costs: The costs of executing,        electronically booking, clearing, and settling derivatives        transactions can be large, sometimes requiring analytical and        database software systems and personnel knowledgeable in such        transactions. While a goal of many in the securities processing        industry is to achieve “straight-through-processing” of        derivatives transactions, many derivatives counterparties        continue to manage the processing of these transactions using a        combination of electronic and manual steps which are not        particularly integrated or automated and therefore add to costs.    -   (6) Event Risk: Most traders understand effective hedging of        derivatives transactions to require markets to be liquid and to        exhibit continuously fluctuating prices without sudden and        dramatic “gaps.” During periods of financial crises and        disequilibria, it is not uncommon to observe dramatic repricing        of underlying securities by 50% or more in a period of hours.        The event risk of such crises and disequilibria are therefore        customarily factored into derivatives prices by dealers, which        increases the cost of derivatives in excess of the theoretical        prices indicated by derivatives valuation models. These costs        are usually spread across all derivatives users.    -   (7) Model Risk: Derivatives contracts can be quite difficult to        value, especially those involving interest rates or features        which allow a counterparty to make decisions throughout the life        of the derivative (e.g., American options allow a counterparty        to realize the value of the derivative at any time during its        life). Derivatives dealers will typically add a premium to        derivatives prices to insure against the possibility that the        valuation models may not adequately reflect market factors or        other conditions throughout the life of the contract. In        addition, risk management guidelines may require firms to        maintain additional capital supporting a derivatives dealing        operation where model risk is determined to be a significant        factor. Model risk has also been a large factor in well-known        cases where complicated securities risk management systems have        provided incorrect or incomplete information, such as the Joe        Jett/Kidder Peabody losses of 1994.    -   (8) Asymmetric Information: Derivatives dealers and market        makers customarily seek to protect themselves from        counterparties with superior information. Bid-offer spreads for        derivatives therefore usually reflect a built-in insurance        premium for the dealer for transactions with counterparties with        superior information, which can lead to unprofitable        transactions. Traditional insurance markets also incur costs due        to asymmetric information. In property-casualty lines, the        direct writer of the insurance almost always has superior        information regarding the book of risks than does the assuming        reinsurer. Much like the market maker in capital markets, the        reinsurer typically prices its informational disadvantage into        the reinsurance premiums.    -   (9) Incomplete Markets: Traditional capital and insurance        markets are often viewed as incomplete in the sense that the        span of contingent claims is limited, i.e., the markets may not        provide opportunities to hedge all of the risks for which        hedging opportunities are sought. As a consequence, participants        typically either bear risk inefficiently or use less than        optimal means to transfer or hedge against risk. For example,        the demand by some investors to hedge inflation risk has        resulted in the issuance by some governments of inflation-linked        bonds which have coupons and principal amounts linked to        Consumer Price Index (CPI) levels. This provides a degree of        insurance against inflation risk. However, holders of such bonds        frequently make assumptions as to the future relationship        between real and nominal interest rates. An imperfect        correlation between the contingent claim (in this case,        inflation-linked bond) and the contingent event (inflation)        gives rise to what traders call “basis risk,” which is risk        that, in today's markets, cannot be perfectly insured or hedged.

Currently, transaction costs are also considerable in traditionalinsurance and reinsurance markets. In recent years, considerable efforthas been expended in attempting to securitize insurance risk such asproperty-casualty catastrophe risk. Traditional insurance andreinsurance markets in many respects resemble principal market-makersecurities markets and suffer from many of the same shortcomings andincur similar costs of operation. Typically, risk is physicallytransferred contractually, credit status of counterparties is monitored,and sophisticated risk management systems are deployed and maintained.Capitalization levels to support insurance portfolios of risky assetsand liabilities may be dramatically out of equilibrium at any given timedue to price stickiness, informational asymmetries and costs, andregulatory constraints. In short, the insurance and reinsurance marketstend to operate according to the same market mechanisms that haveprevailed for decades, despite large market shocks such as the Lloydscrisis in the late 1980's and early 1990's.

Accordingly, a driving force behind all the contributors to the costs ofderivatives and insurance contracts is the necessity or desirability ofrisk management through dynamic hedging or contingent claim replicationin continuous, liquid, and informationally fair markets. Hedging is usedby derivatives dealers to reduce their exposure to excessive market riskwhile making transaction fees to cover their cost of capital and ongoingoperations; and effective hedging requires liquidity.

Recent patents have addressed the problem of financial market liquidityin the context of an electronic order-matching systems (e.g., U.S. Pat.No. 5,845,266). The principal techniques disclosed to enhance liquidityare to increase participation and traded volume in the system and tosolicit trader preferences about combinations of price and quantity fora particular trade of a security. There are shortcomings to thesetechniques, however. First, these techniques implement order-matchingand limit order book algorithms, which can be and are effectivelyemployed in traditional “brick and mortar” exchanges. Their electronicimplementation, however, primarily serves to save on transportation andtelecommunication charges. No fundamental change is contemplated tomarket structure for which an electronic network may be essential.Second, the disclosed techniques appear to enhance liquidity at theexpense of placing large informational burdens on the traders (bysoliciting preferences, for example, over an entire price-quantitydemand curve) and by introducing uncertainty as to the exact price atwhich a trade has been transacted or is “filled.” Finally, theseelectronic order matching systems contemplate a traditional counterpartypairing, which means physical securities are frequently transferred,cleared, and settled after the counterparties are identified andmatched. In other words, techniques disclosed in the context ofelectronic order-matching systems are technical elaborations to thebasic problem of how to optimize the process of matching arrays of bidsand offers.

Patents relating to derivatives, such as U.S. Pat. No. 4,903,201,disclose an electronic adaptation of current open-outcry or ordermatching exchanges for the trading of futures is disclosed. Anotherrecent patent, U.S. Pat. No. 5,806,048, relates to the creation ofopen-end mutual fund derivative securities to provide enhanced liquidityand improved availability of information affecting pricing. This patent,however, does not contemplate an electronic derivatives exchange whichrequires the traditional hedging or replicating portfolio approach tosynthesizing the financial derivatives. Similarly, U.S. Pat. No.5,794,207 proposes an electronic means of matching buyers' bids andsellers' offers, without explaining the nature of the economic priceequilibria achieved through such a market process.

SUMMARY OF THE INVENTION

The present invention is directed to systems and methods of trading, andfinancial products, having a goal of reducing transaction costs formarket participants who hedge against or otherwise make investments incontingent claims relating to events of economic significance. Theclaims are contingent in that their payout or return depends on theoutcome of an observable event with more than one possible outcome. Anexample of such a contingent claim is a digital option, such as adigital call option, where the investor receives a payout if theunderlying asset, stock or index expires at or above a specified strikeprice and receives no payout if the underlying asset, stock or otherindex expires below the strike price. Digital options can also bereferred to as, for example, “binary options” and “all or nothingoptions.” The contingent claims relate to events of economicsignificance in that an investor or trader in a contingent claimtypically is not economically indifferent to the outcome of the event,even if the investor or trader has not invested in or traded acontingent claim relating to the event.

Intended users of preferred and other embodiments of the presentinvention are typically institutional investors, such as financialinstitutions including banks, investment banks, primary insurers andreinsurers, and corporate treasurers, hedge funds and pension funds.Users can also include any individual or entity with a need for riskallocation services. As used in this specification, the terms “user,”“trader” and “investor” are used interchangeably to mean anyinstitution, individual or entity that desires to trade or invest incontingent claims or other financial products described in thisspecification.

The contingent claims pertaining to an event have a trading period or anauction period in order to finalize a return for each defined state,each defined state corresponding to an outcome or set of outcomes forthe event, and another period for observing the event upon which thecontingent claim is based. When the contingent claim is a digitaloption, the price or investment amount for each digital option isfinalized at the end of the trading period, along with the return foreach defined state. The entirety of trades or orders placed and acceptedwith respect to a certain trading period are processed in a demand-basedmarket or auction. The organization or institution, individual or otherentity sponsoring, running, maintaining or operating the demand-basedmarket or auction, can be referred to, for example, as an “exchange,”“auction sponsor” and/or “market sponsor.”

In each market or auction, the returns to the contingent claims adjustduring the trading period of the market or auction with changes in thedistribution of amounts invested in each of the states. The investmentamounts for the contingent claims can either be provided up front ordetermined during the trading period with changes in the distribution ofdesired returns and selected outcomes for each claim. The returnspayable for each of the states are finalized after the conclusion ofeach relevant trading period. In a preferred embodiment, the totalamount invested, less a transaction fee to an exchange, or a market orauction sponsor, is equal to the total amount of the payouts. In otherwords, in theory, the returns on all of the contingent claimsestablished during a particular trading period and pertaining to aparticular event are essentially zero sum, as are the traditionalderivatives markets. In one embodiment, the investment amounts or pricesfor each contingent claim are finalized after the conclusion of eachrelevant trading period, along with the returns payable for each of thestates. Since the total amount invested, less a transaction fee to anexchange, or a market or auction sponsor, is equal to the total amountof payouts, an optimization solution using an iteration algorithmdescribed below can be used to determine the equilibrium investmentamounts or prices for each contingent claim along with establishing thereturns on all of the contingent claims, given the desired or requestedreturn for each claim, the selection of outcomes for each claim and thelimit (if any) on the investment amount for each claim.

The process by which returns and investment amounts for each contingentclaim are finalized in the present invention is demand-based, and doesnot in any substantial way depend on supply. By contrast, traditionalmarkets set prices through the interaction of supply and demand bycrossing bids to buy and offers to sell (“bid/offer”). The demand-basedcontingent claim mechanism of the present invention sets returns byfinancing returns to successful investments with losses fromunsuccessful investments. Thus, in a preferred embodiment, the returnsto successful investments (as well as the prices or investment amountsfor investments in digital options) are determined by the total andrelative amounts of all investments placed on each of the defined statesfor the specified observable event.

As used in this specification, the term “contingent claim” shall havethe meaning customarily ascribed to it in the securities, trading,insurance and economics communities. “Contingent claims” thus include,for example, stocks, bonds and other such securities, derivativesecurities, insurance contracts and reinsurance agreements, and anyother financial products, instruments, contracts, assets, or liabilitieswhose value depends upon or reflects economic risk due to the occurrenceof future, real-world events. These events may be financial-relatedevents, such as changes in interest rates, or non-financial-relatedevents such as changes in weather conditions, demand for electricity,and fluctuations in real estate prices. Contingent claims also includeall economic or financial interests, whether already traded or not yettraded, which have or reflect inherent risk or uncertainty due to theoccurrence of future real-world events. Examples of contingent claims ofeconomic or financial interest which are not yet traded on traditionalmarkets are financial products having values that vary with thefluctuations in corporate earnings or changes in real estate values andrentals. The term “contingent claim” as used in this specificationencompasses both hypothetical financial products of the Arrow-Debreuvariety, as well as any risky asset, contract or product which can beexpressed as a combination or portfolio of the hypothetical financialproducts.

For the purposes of this specification, an “investment” in or “trade” oran “order” of a contingent claim is the act of putting an amount (in theunits of value defined by the contingent claim) at risk, with afinancial return depending on the outcome of an event of economicsignificance underlying the group of contingent claims pertaining tothat event.

“Derivative security” (used interchangeably with “derivative”) also hasa meaning customarily ascribed to it in the securities, trading,insurance and economics communities. This includes a security orcontract whose value depends on such factors as the value of anunderlying security, index, asset or liability, or on a feature of suchan underlying security, such as interest rates or convertibility intosome other security. A derivative security is one example of acontingent claim as defined above. Financial futures on stock indicessuch as the S&P 500 or options to buy and sell such futures contractsare highly popular exchange-traded financial derivatives. Aninterest-rate swap, which is an example of an off-exchange derivative,is an agreement between two counterparties to exchange series ofcashflows based on underlying factors, such as the London InterbankOffered Rate (LIBOR) quoted daily in London for a large number offoreign currencies. Like the exchange-traded futures and options,off-exchange agreements can fluctuate in value with the underlyingfactors to which they are linked or derived. Derivatives may also betraded on commodities, insurance events, and other events, such as theweather.

In this specification, the function for computing and allocating returnsto contingent claims is termed the Demand Reallocation Function (DRF). ADRF is demand-based and involves reallocating returns to investments ineach state after the outcome of the observable event is known in orderto compensate successful investments from losses on unsuccessfulinvestments (after any transaction or exchange fee). Since an adjustablereturn based on variations in amounts invested is a key aspect of theinvention, contingent claims implemented using a DRF will be referred toas demand-based adjustable return (DBAR) contingent claims.

In accordance with embodiments of the present invention, an Order PriceFunction (OPF) is a function for computing the investment amounts orprices for contingent claims which are digital options. An OPF, whichincludes the DRF, is also demand-based and involves determining theprices for each digital option at the end of the trading period, butbefore the outcome of the observable event is known. The OPF determinesthe prices as a function of the outcomes selected in each digital option(corresponding to the states selected by a trader for the digital optionto be in-the-money), the requested payout for the digital option if theoption expires in-the money, and the limit placed on the price (if any)when the order for the option is placed in the market or auction.

“Demand-based market,” “demand-based auction” may include, for example,a market or auction which is run or executed according to the principlesset forth in the embodiments of the present invention. “Demand-basedtechnology” may include, for example, technology used to run or executeorders in a demand-based market or auction in accordance with theprinciples set forth in the embodiments of the present invention.“Contingent claims” or “DBAR contingent claims” may include, forexample, contingent claims that are processed in a demand-based marketor auction. “Contingent claims” or “DBAR contingent claims” may include,for example, digital options or DBAR digital options, discussed in thisspecification. With respect to digital options, demand-based markets mayinclude, for example, DBAR DOEs (DBAR Digital Option Exchanges), orexchanges in which orders for digital options or DBAR digital optionsare placed and processed. “Contingent claims” or “DBAR contingentclaims” may also include, for example, DBAR-enabled products orDBAR-enabled financial products, discussed in this specification.

Preferred features of a trading system for a group of DBAR contingentclaims (i.e., group of claims pertaining to the same event) include thefollowing: (1) an entire distribution of states is open for investment,not just a single price as in the traditional markets; (2) returns areadjustable and determined mathematically based on invested amounts ineach of the states available for investment, (3) invested amounts arepreferably non-decreasing (as explained below), providing a commitmentof offered liquidity to the market over the distribution of states, andin one embodiment of the present invention, adjustable and determinedmathematically based on requested returns per order, selection ofoutcomes for the option to expire in-the-money, and limit amounts (ifany), and (4) information is available in real-time across thedistribution of states, including, in particular, information on theamounts invested across the distribution of all states (commonly knownas a “limit order book”). Other consequences of preferred embodiments ofthe present invention include (1) elimination of order-matching orcrossing of the bid and offer sides of the market; (2) reduction of theneed for a market maker to conduct dynamic hedging and risk management;(3) more opportunities for hedging and insuring events of economicsignificance (i.e., greater market “completeness”); and (4) the abilityto offer investments in contingent claims whose profit and lossscenarios are comparable to these for digital options or otherderivatives in traditional markets, but can be implemented using theDBAR systems and methods of the present invention, for example withoutthe need for sellers of such options or derivatives as they function inconventional markets.

Other preferred embodiments of the present invention can accommodaterealization of profits and losses by traders at multiple points beforeall of the criteria for terminating a group of contingent claims areknown. This is accomplished by arranging a plurality of trading periods,each having its own set of finalized returns. Profit or loss can berealized or “locked-in” at the end of each trading period, as opposed towaiting for the final outcome of the event on which the relevantcontingent claims are based. Such lock-in can be achieved by placinghedging investments in successive trading periods as the returns change,or adjust, from period to period. In this way, profit and loss can berealized on an evolving basis (limited only by the frequency and lengthof the periods), enabling traders to achieve the same or perhaps higherfrequency of trading and hedging than available in traditional markets.

If desired, an issuer such as a corporation, investment bank,underwriter or other financial intermediary can create a security havingreturns that are driven in a comparable manner to the DBAR contingentclaims of the present invention. For example, a corporation may issue abond with returns that are linked to insurance risk. The issuer cansolicit trading and calculate the returns based on the amounts investedin contingent claims corresponding to each level or state of insurancerisks.

In a preferred embodiment of the present invention, changes in thereturn for investments in one state will affect the return oninvestments in another state in the same distribution of states for agroup of contingent claims. Thus, traders' returns will depend not onlyon the actual outcome of a real-world, observable event but also ontrading choices from among the distribution of states made by othertraders. This aspect of DBAR markets, in which returns for one state areaffected by changes in investments in another state in the samedistribution, allows for the elimination of order-crossing and dynamicmarket maker hedging. Price-discovery in preferred embodiments of thepresent invention can be supported by a one-way market (i.e., demand,not supply) for DBAR contingent claims. By structuring derivatives andinsurance trading according to DBAR principles, the high costs oftraditional order matching and principal market making market structurescan be reduced substantially. Additionally, a market implemented bysystems and methods of the present invention is especially amenable toelectronic operation over a wide network, such as the Internet.

In its preferred embodiments, the present invention mitigatesderivatives transaction costs found in traditional markets due todynamic hedging and order matching. A preferred embodiment of thepresent invention provides a system for trading contingent claimsstructured under DBAR principles, in which amounts invested in on eachstate in a group of DBAR contingent claims are reallocated fromunsuccessful investments, under defined rules, to successful investmentsafter the deduction of exchange transaction fees. In particular, theoperator of such a system or exchange provides the physical plant andelectronic infrastructure for trading to be conducted, collects andaggregates investments (or in one embodiment, first collects andaggregates investment information to determine investment amounts pertrade or order and then collects and aggregates the investment amounts),calculates the returns that result from such investments, and thenallocates to the successful investments returns that are financed by theunsuccessful investments, after deducting a transaction fee for theoperation of the system.

In preferred embodiments, where the successful investments are financedwith the losses from unsuccessful investments, returns on all trades arecorrelated and traders make investments against each other as well asassuming the risk of chance outcomes. All traders for a group of DBARcontingent claims depending on a given event become counterparties toeach other, leading to a mutualization of financial interests.Furthermore, in preferred embodiments of the present invention,projected returns prevailing at the time an investment is made may notbe the same as the final payouts or returns after the outcome of therelevant event is known.

Traditional derivatives markets by contrast, operate largely under ahouse “banking” system. In this system, the market-maker, whichtypically has the function of matching buyers and sellers, customarilyquotes a price at which an investor may buy or sell. If a given investorbuys or sells at the price, the investor's ultimate return is based uponthis price, i.e., the price at which the investor later sells or buysthe original position, along with the original price at which theposition was traded, will determine the investor's return. As themarket-maker may not be able perfectly to offset buy and sell orders atall times or may desire to maintain a degree of risk in the expectationof returns, it will frequently be subject to varying degrees of marketrisk (as well as credit risk, in some cases). In a traditionalderivatives market, market-makers which match buy and sell orderstypically rely upon actuarial advantage, bid-offer spreads, a largecapital base, and “coppering” or hedging (risk management) to minimizethe chance of bankruptcy due to such market risk exposures.

Each trader in a house banking system typically has only a singlecounterparty—the market-maker, exchange, or trading counterparty (in thecase, for example, of over-the-counter derivatives). By contrast,because a market in DBAR contingent claims may operate according toprinciples whereby unsuccessful investments finance the returns onsuccessful investments, the exchange itself is exposed to reduced riskof loss and therefore has reduced need to transact in the market tohedge itself. In preferred embodiments of DBAR contingent claims of thepresent invention, dynamic hedging or bid-offer crossing by the exchangeis generally not required, and the probability of the exchange ormarket-maker going bankrupt may be reduced essentially to zero. Such asystem distributes the risk of bankruptcy away from the exchange ormarket-maker and among all the traders in the system. The system as awhole provides a great degree of self-hedging and substantial reductionof the risk of market failure for reasons related to market risk. A DBARcontingent claim exchange or market or auction may also be“self-clearing” and require little clearing infrastructure (such asclearing agents, custodians, nostro/vostro bank accounts, and transferand register agents). A derivatives trading system or exchange or marketor auction structured according to DBAR contingent claim principlestherefore offers many advantages over current derivatives marketsgoverned by house banking principles.

The present invention also differs from electronic or parimutuel bettingsystems disclosed in the prior art (e.g., U.S. Pat. Nos. 5,873,782 and5,749,785). In betting systems or games of chance, in the absence of awager the bettor is economically indifferent to the outcome (assumingthe bettor does not own the casino or the racetrack or breed the racinghorses, for example). The difference between games of chance and eventsof economic significance is well known and understood in financialmarkets.

In summary, the present invention provides systems and methods forconducting demand-based trading. A preferred embodiment of a method ofthe present invention for conducting demand-based trading includes thesteps of (a) establishing a plurality of defined states and plurality ofpredetermined termination criteria, wherein each of the defined statescorresponds to at least one possible outcome of an event of economicsignificance; (b) accepting investments of value units by a plurality oftraders in the defined states; and (c) allocating a payout to eachinvestment. The allocating step is responsive to the total number ofvalue units invested in the defined states, the relative number of valueunits invested in each of the defined states, and the identification ofthe defined state that occurred upon fulfillment of all of thetermination criteria.

An additional preferred embodiment of a method for conductingdemand-based trading also includes establishing, accepting, andallocating steps. The establishing step in this embodiment includesestablishing a plurality of defined states and a plurality ofpredetermined termination criteria. Each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled. The accepting stepincludes accepting investments of value units by multiple traders in thedefined states. The allocating step includes allocating a payout to eachinvestment. This allocating step is responsive to the total number ofvalue units invested in the defined states, the relative number of valueunits invested in each of the defined states, and the identification ofthe defined state that occurred upon fulfillment of all of thetermination criteria.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, the payout to each investment in each of thedefined states that did not occur upon fulfillment of all of thetermination criteria is zero, and the sum of the payouts to all of theinvestments is not greater than the value of the total number of thevalue units invested in the defined states. In a further preferredembodiment, the sum of the values of the payouts to all of theinvestments is equal to the value of all of the value units invested indefined states, less a fee.

In preferred embodiments of a method for conducting demand-basedtrading, at least one investment of value units designates a set ofdefined states and a desired return-on-investment from the designatedset of defined states. In these preferred embodiments, the allocatingstep is further responsive to the desired return-on-investment from thedesignated set of defined states.

In another preferred embodiment of a method for conducting demand-basedtrading, the method further includes the step of calculatingCapital-At-Risk for at least one investment of value units by at leastone trader. In alternative further preferred embodiments, the step ofcalculating Capital-At-Risk includes the use of the Capital-At-RiskValue-At-Risk method, the Capital-At-Risk Monte Carlo Simulation method,or the Capital-At-Risk Historical Simulation method.

In preferred embodiments of a method for conducting demand-basedtrading, the method further includes the step of calculatingCredit-Capital-At-Risk for at least one investment of value units by atleast one trader. In alternative further preferred embodiments, the stepof calculating Credit-Capital-At-Risk includes the use of theCredit-Capital-At-Risk Value-At-Risk method, the Credit-Capital-At-RiskMonte Carlo Simulation method, or the Credit-Capital-At-Risk HistoricalSimulation method.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, at least one investment of value units is amulti-state investment that designates a set of defined states. In afurther preferred embodiment, at least one multi-state investmentdesignates a set of desired returns that is responsive to the designatedset of defined states, and the allocating step is further responsive tothe set of desired returns. In a further preferred embodiment, eachdesired return of the set of desired returns is responsive to a subsetof the designated set of defined states. In an alternative preferredembodiment, the set of desired returns approximately corresponds toexpected returns from a set of defined states of a prespecifiedinvestment vehicle such as, for example, a particular call option.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, the allocating step includes the steps of (a)calculating the required number of value units of the multi-stateinvestment that designates a set of desired returns, and (b)distributing the value units of the multi-state investment thatdesignates a set of desired returns to the plurality of defined states.In a further preferred embodiment, the allocating step includes the stepof solving a set of simultaneous equations that relate traded amounts tounit payouts and payout distributions; and the calculating step and thedistributing step are responsive to the solving step.

In preferred embodiment's of a method for conducting demand-basedtrading of the present invention, the solving step includes the step offixed point iteration. In further preferred embodiments, the step offixed point iteration includes the steps of (a) selecting an equation ofthe set of simultaneous equations described above, the equation havingan independent variable and at least one dependent variable; (b)assigning arbitrary values to each of the dependent variables in theselected equation; (c) calculating the value of the independent variablein the selected equation responsive to the currently assigned values ofeach the dependent variables; (d) assigning the calculated value of theindependent variable to the independent variable; (e) designating anequation of the set of simultaneous equations as the selected equation;and (f) sequentially performing the calculating the value step, theassigning the calculated value step, and the designating an equationstep until the value of each of the variables converges.

A preferred embodiment of a method for estimating state probabilities ina demand-based trading method of the present invention includes thesteps of: (a) performing a demand-based trading method having aplurality of defined states and a plurality of predetermined terminationcriteria, wherein an investment of value units by each of a plurality oftraders is accepted in at least one of the defined states, and at leastone of these defined states corresponds to at least one possible outcomeof an event of economic significance; (b) monitoring the relative numberof value units invested in each of the defined states; and (c)estimating, responsive to the monitoring step, the probability that aselected defined state will be the defined state that occurs uponfulfillment of all of the termination criteria.

An additional preferred embodiment of a method for estimating stateprobabilities in a demand-based trading method also includes performing,monitoring, and estimating steps. The performing step includesperforming a demand-based trading method having a plurality of definedstates and a plurality of predetermined termination criteria, wherein aninvestment of value units by each of a plurality of traders is acceptedin at least one of the defined states; and wherein each of the definedstates corresponds to a possible state of a selected financial productwhen each of the termination criteria is fulfilled. The monitoring stepincludes monitoring the relative number of value units invested in eachof the defined states. The estimating step includes estimating,responsive to the monitoring step, the probability that a selecteddefined state will be the defined state that occurs upon fulfillment ofall of the termination criteria.

A preferred embodiment of a method for promoting liquidity in ademand-based trading method of the present invention includes the stepof performing a demand-based trading method having a plurality ofdefined states and a plurality of predetermined termination criteria,wherein an investment of value units by each of a plurality of tradersis accepted in at least one of the defined states and wherein anyinvestment of value units cannot be withdrawn after acceptance. Each ofthe defined states corresponds to at least one possible outcome of anevent of economic significance. A further preferred embodiment of amethod for promoting liquidity in a demand-based trading method includesthe step of hedging. The hedging step includes the hedging of a trader'sprevious investment of value units by making a new investment of valueunits in one or more of the defined states not invested in by theprevious investment.

An additional preferred embodiment of a method for promoting liquidityin a demand-based trading method includes the step of performing ademand-based trading method having a plurality of defined states and aplurality of predetermined termination criteria, wherein an investmentof value units by each of a plurality of traders is accepted in at leastone of the defined states and wherein any investment of value unitscannot be withdrawn after acceptance, and each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled. A further preferredembodiment of such a method for promoting liquidity in a demand-basedtrading method includes the step of hedging. The hedging step includesthe hedging of a trader's previous investment of value units by making anew investment of value units in one or more of the defined states notinvested in by the previous investment.

A preferred embodiment of a method for conducting quasi-continuousdemand-based trading includes the steps of: (a) establishing a pluralityof defined states and a plurality of predetermined termination criteria,wherein each of the defined states corresponds to at least one possibleoutcome of an event; (b) conducting a plurality of trading cycles,wherein each trading cycle includes the step of accepting, during apredefined trading period and prior to the fulfillment of all of thetermination criteria, an investment of value units by each of aplurality of traders in at least one of the defined states; and (c)allocating a payout to each investment. The allocating step isresponsive to the total number of the value units invested in thedefined states during each of the trading periods, the relative numberof the value units invested in each of the defined states during each ofthe trading periods, and an identification of the defined state thatoccurred upon fulfillment of all of the termination criteria. In afurther preferred embodiment of a method for conducting quasi-continuousdemand-based trading, the predefined trading periods are sequential anddo not overlap.

Another preferred embodiment of a method for conducting demand-basedtrading includes the steps of: (a) establishing a plurality of definedstates and a plurality of predetermined termination criteria, whereineach of the defined states corresponds to one possible outcome of anevent of economic significance (or a financial instrument); (b)accepting, prior to fulfillment of all of the termination criteria, aninvestment of value units by each of a plurality of traders in at leastone of the plurality of defined states, with at least one investmentdesignating a range of possible outcomes corresponding to a set ofdefined states; and (c) allocating a payout to each investment. In sucha preferred embodiment, the allocating step is responsive to the totalnumber of value units in the plurality of defined states, the relativenumber of value units invested in each of the defined states, and anidentification of the defined state that occurred upon the fulfillmentof all of the termination criteria. Also in such a preferred embodiment,the allocation is done so that substantially the same payout isallocated to each state of the set of defined states. This embodimentcontemplates, among other implementations, a market or exchange forcontingent claims of the present invention that provides—withouttraditional sellers—profit and loss scenarios comparable to thoseexpected by traders in derivative securities known as digital options,where payout is the same if the option expires anywhere in the money,and where there is no payout if the option expires out of the money.

Another preferred embodiment of the present invention provides a methodfor conducting demand-based trading including: (a) establishing aplurality of defined states and a plurality of predetermined terminationcriteria, wherein each of the defined states corresponds to one possibleoutcome of an event of economic significance (or a financialinstrument); (b) accepting, prior to fulfillment of all of thetermination criteria, a conditional investment order by a trader in atleast one of the plurality of defined states; (c) computing, prior tofulfillment of all of the termination criteria a probabilitycorresponding to each defined state; and (d) executing or withdrawing,prior to the fulfillment of all of the termination criteria, theconditional investment responsive to the computing step. In suchembodiments, the computing step is responsive to the total number ofvalue units invested in the plurality of defined states and the relativenumber of value units invested in each of the plurality of definedstates. Such embodiments contemplate, among other implementations, amarket or exchange (again without traditional sellers) in whichinvestors can make and execute conditional or limit orders, where anorder is executed or withdrawn in response to a calculation of aprobability of the occurrence of one or more of the defined states.Preferred embodiments of the system of the present invention involve theuse of electronic technologies, such as computers, computerizeddatabases and telecommunications systems, to implement methods forconducting demand-based trading of the present invention.

A preferred embodiment of a system of the present invention forconducting demand-based trading includes (a) means for accepting, priorto the fulfillment of all predetermined termination criteria,investments of value units by a plurality of traders in at least one ofa plurality of defined states, wherein each of the defined statescorresponds to at least one possible outcome of an event of economicsignificance; and (b) means for allocating a payout to each investment.This allocation is responsive to the total number of value unitsinvested in the defined states, the relative number of value unitsinvested in each of the defined states, and the identification of thedefined state that occurred upon fulfillment of all of the terminationcriteria.

An additional preferred embodiment of a system of the present inventionfor conducting demand-based trading includes (a) means for accepting,prior to the fulfillment of all predetermined termination criteria,investments of value units by a plurality of traders in at least one ofa plurality of defined states, wherein each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled; and (b) means forallocating a payout to each investment. This allocation is responsive tothe total number of value units invested in the defined states, therelative number of value units invested in each of the defined states,and the identification of the defined state that occurred uponfulfillment of all of the termination criteria.

A preferred embodiment of a demand-based trading apparatus of thepresent invention includes (a) an interface processor communicating witha plurality of traders and a market data system; and (b) a demand-basedtransaction processor, communicating with the interface processor andhaving a trade status database. The demand-based transaction processormaintains, responsive to the market data system and to a demand-basedtransaction with one of the plurality of traders, the trade statusdatabase, and processes, responsive to the trade status database, thedemand-based transaction.

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, maintaining the trade status database includes(a) establishing a contingent claim having a plurality of definedstates, a plurality of predetermined termination criteria, and at leastone trading period, wherein each of the defined states corresponds to atleast one possible outcome of an event of economic significance; (b)recording, responsive to the demand-based transaction, an investment ofvalue units by one of the plurality of traders in at least one of theplurality of defined states; (c) calculating, responsive to the totalnumber of the value units invested in the plurality of defined statesduring each trading period and responsive to the relative number of thevalue units invested in each of the plurality of defined states duringeach trading period, finalized returns at the end of each tradingperiod; and (d) determining, responsive to an identification of thedefined state that occurred upon the fulfillment of all of thetermination criteria and to the finalized returns, payouts to each ofthe plurality of traders; and processing the demand-based transactionincludes accepting, during the trading period, the investment of valueunits by one of the plurality of traders in at least one of theplurality of defined states;

In an alternative further preferred embodiment of a demand-based tradingapparatus of the present invention, maintaining the trade statusdatabase includes (a) establishing a contingent claim having a pluralityof defined states, a plurality of predetermined termination criteria,and at least one trading period, wherein each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled; (b) recording, responsiveto the demand-based transaction, an investment of value units by one ofthe plurality of traders in at least one of the plurality of definedstates; (c) calculating, responsive to the total number of the valueunits invested in the plurality of defined states during each tradingperiod and responsive to the relative number of the value units investedin each of the plurality of defined states during each trading period,finalized returns at the end of each trading period; and (d)determining, responsive to an identification of the defined state thatoccurred upon the fulfillment of all of the termination criteria and tothe finalized returns, payouts to each of the plurality of traders; andprocessing the demand-based transaction includes accepting, during thetrading period, the investment of value units by one of the plurality oftraders in at least one of the plurality of defined states;

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, maintaining the trade status database includescalculating return estimates; and processing the demand-basedtransaction includes providing, responsive to the demand-basedtransaction, the return estimates.

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, maintaining the trade status database includescalculating risk estimates; and processing the demand-based transactionincludes providing, responsive to the demand-based transaction, the riskestimates.

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, the demand-based transaction includes amulti-state investment that specifies a desired payout distribution anda set of constituent states; and maintaining the trade status databaseincludes allocating, responsive to the multi-state investment, valueunits to the set of constituent states to create the desired payoutdistribution. Such demand-based transactions may also includemulti-state investments that specify the same payout if any of adesignated set of states occurs upon fulfillment of the terminationcriteria. Other demand-based transactions executed by the demand-basedtrading apparatus of the present invention include conditionalinvestments in one or more states, where the investment is executed orwithdrawn in response to a calculation of a probability of theoccurrence of one or more states upon the fulfillment of the terminationcriteria.

In an additional embodiment, systems and methods for conductingdemand-based trading includes the steps of (a) establishing a pluralityof states, each state corresponding to at least one possible outcome ofan event of economic significance; (b) receiving an indication of adesired payout and an indication of a selected outcome, the selectedoutcome corresponding to at least one of the plurality of states; and(c) determining an investment amount as a function of the selectedoutcome, the desired payout and a total amount invested in the pluralityof states.

In another additional embodiment, systems and methods for conductingdemand-based trading includes the steps of (a) establishing a pluralityof states, each state corresponding to at least one possible outcome ofan event (whether or not such event is an economic event); (b) receivingan indication of a desired payout and an indication of a selectedoutcome, the selected outcome corresponding to at least one of theplurality of states; and (c) determining an investment amount as afunction of the selected outcome, the desired payout and a total amountinvested in the plurality of states.

In another additional embodiment, systems and methods for conductingdemand-based trading includes the steps of (a) establishing a pluralityof states, each state corresponding to at least one possible outcome ofan event of economic significance; (b) receiving an indication of aninvestment amount and a selected outcome, the selected outcomecorresponding to at least one of the plurality of states; and (c)determining a payout as a function of the investment amount, theselected outcome, a total amount invested in the plurality of states,and an identification of at least one state corresponding to an observedoutcome of the event.

In another additional embodiment, systems and methods for conductingdemand-based trading include the steps of: (a) receiving an indicationof one or more parameters of a financial product; and (b) determiningone or more of a selected outcome, a desired payout, an investmentamount, and a limit on the investment amount for each contingent claimin a set of one or more contingent claims as a function of the one ormore financial product parameters.

In another additional embodiment, systems and methods for conductingdemand-based trading include the steps of: (a) receiving an indicationof one or more parameters of a financial product; and (b) determining aninvestment amount and a selected outcome for each contingent claim in aset of one or more contingent claims as a function of the one or morefinancial product parameters.

In another additional embodiment, a demand-enabled financial product fortrading in a demand-based auction includes a set of one or morecontingent claims, the set approximating a financial product, eachcontingent claim in the set having an investment amount and a selectedoutcome, each investment amount being dependent upon one or moreparameters of a financial product and a total amount invested in theauction.

An object of the present invention is to provide systems and methods tosupport and facilitate a market structure for contingent claims relatedto observable events of economic significance, which includes one ormore of the following advantages, in addition to those described above:

-   -   1. ready implementation and support using electronic computing        and networking technologies;    -   2. reduction or elimination of the need to match bids to buy        with offers to sell in order to create a market for derivatives;    -   3. reduction or elimination of the need for a derivatives        intermediary to match bids and offers;    -   4. mathematical and consistent calculation of returns based on        demand for contingent claims;    -   5. increased liquidity and liquidity incentives;    -   6. statistical diversification of credit risk through the        mutualization of multiple derivatives counterparties;    -   7. improved scalability by reducing the traditional linkage        between the method of pricing for contingent claims and the        quantity of the underlying claims available for investment;    -   8. increased price transparency;    -   9. improved efficiency of information aggregation mechanisms;    -   10. reduction of event risk, such as the risk of discontinuous        market events such as crashes;    -   11. opportunities for binding offers of liquidity to the market;    -   12. reduced incentives for strategic behavior by traders;    -   13. increased market for contingent claims;    -   14. improved price discovery;    -   15. improved self-consistency;    -   16. reduced influence by market makers;    -   17. ability to accommodate virtually unlimited demand;    -   18. ability to isolate risk exposures;    -   19. increased trading precision, transaction certainty and        flexibility;    -   20. ability to create valuable new markets with a sustainable        competitive advantage;    -   21. new source of fee revenue without putting capital at risk;        and    -   22. increased capital efficiency.

A further object of the present invention is to provide systems andmethods for the electronic exchange of contingent claims related toobservable events of economic significance, which includes one or moreof the following advantages:

-   -   1. reduced transaction costs, including settlement and clearing        costs, associated with derivatives transactions and insurable        claims;    -   2. reduced dependence on complicated valuation models for        trading and risk management of derivatives;    -   3. reduced need for an exchange or market maker to manage market        risk by hedging;    -   4. increased availability to traders of accurate and up-to-date        information on the trading of contingent claims, including        information regarding the aggregate amounts invested across all        states of events of economic significance, and including over        varying time periods;    -   5. reduced exposure of the exchange to credit risk;    -   6. increased availability of information on credit risk and        market risk borne by traders of contingent claims;    -   7. increased availability of information on marginal returns        from trades and investments that can be displayed        instantaneously after the returns adjust during a trading        period;    -   8. reduced need for a derivatives intermediary or exchange to        match bids and offers;    -   9. increased ability to customize demand-based adjustable return        (DBAR) payouts to permit replication of traditional financial        products and their derivatives;    -   10. comparability of profit and loss scenarios to those expected        by traders for purchases and sales of digital options and other        derivatives, without conventional sellers;    -   11. increased data generation; and    -   12. reduced exposure of the exchange to market risk.

Additional objects and advantages of the invention are set forth in partin the description which follows, and in part are obvious from thedescription, or may be learned by practice of the invention. The objectsand advantages of the invention may also be realized and attained bymeans of the instrumentalities, systems, methods and steps set forth inthe appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and from a part ofthe specification, illustrate embodiments of the present invention and,together with the description, serve to explain the principles of theinvention.

FIG. 1 is a schematic view of various forms of telecommunicationsbetween DBAR trader clients and a preferred embodiment of a DBARcontingent claims exchange implementing the present invention.

FIG. 2 is a schematic view of a central controller of a preferredembodiment of a DBAR contingent claims exchange network architectureimplementing the present invention.

FIG. 3 is a schematic depiction of the trading process on a preferredembodiment of a DBAR contingent claims exchange.

FIG. 4 depicts data storage devices of a preferred embodiment of a DBARcontingent claims exchange.

FIG. 5 is a flow diagram illustrating the processes of a preferredembodiment of DBAR contingent claims exchange in executing a DBAR rangederivatives investment.

FIG. 6 is an illustrative HTML interface page of a preferred embodimentof a DBAR contingent claims exchange.

FIG. 7 is a schematic view of market data flow to a preferred embodimentof a DBAR contingent claims exchange.

FIG. 8 is an illustrative graph of the implied liquidity effects for agroup of DBAR contingent claims.

FIG. 9 a is a schematic representation of a traditional interest rateswap transaction.

FIG. 9 b is a schematic of investor relationships for an illustrativegroup of DBAR contingent claims.

FIG. 9 c shows a tabulation of credit ratings and margin trades for eachinvestor in to an illustrative group of DBAR contingent claims.

FIG. 10 is a schematic view of a feedback process for a preferredembodiment of DBAR contingent claims exchange.

FIG. 11 depicts illustrative DBAR data structures for use in a preferredembodiment of a Demand-Based Adjustable Return Digital Options Exchangeof the present invention.

FIG. 12 depicts a preferred embodiment of a method for processing limitand market orders in a Demand-Based Adjustable Return Digital OptionsExchange of the present invention.

FIG. 13 depicts a preferred embodiment of a method for calculating amultistate composite equilibrium in a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 14 depicts a preferred embodiment of a method for calculating amultistate profile equilibrium in a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 15 depicts a preferred embodiment of a method for converting “sale”orders to buy orders in a Demand-Based Adjustable Return Digital OptionsExchange of the present invention.

FIG. 16: depicts a preferred embodiment of a method for adjustingimplied probabilities for demand-based adjustable return contingentclaims to account for transaction or exchange fees in a Demand-BasedAdjustable Return Digital Options Exchange of the present invention.

FIG. 17 depicts a preferred embodiment of a method for filling andremoving lots of limit orders in a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 18 depicts a preferred embodiment of a method of payoutdistribution and fee collection in a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 19 depicts illustrative DBAR data structures used in anotherembodiment of a Demand-Based Adjustable Return Digital Options Exchangeof the present invention.

FIG. 20 depicts another embodiment of a method for processing limit andmarket orders in another embodiment of a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 21 depicts an upward shift in the earnings expectations curve whichcan be protected by trading digital options and other contingent claimson earnings in successive quarters according to the embodiments of thepresent invention.

FIG. 22 depicts a network implementation of a demand-based market orauction according to the embodiments of the present invention.

FIG. 23 depicts cash flows for each participant trading aprinciple-protected ECI-linked FRN.

FIG. 24 depicts an example time line for a demand-based market tradingDBAR-enabled FRNs or swaps according to the embodiments of the presentinvention.

FIG. 25 depicts an example of an embodiment of a demand-based market orauction with digital options and DBAR-enabled products.

FIG. 26 depicts an example of simple graphical representations ofdigital calls, puts, spreads, and strips.

FIG. 27 depicts the results of a grouping process for a set ofillustrative and assumed digital puts and calls.

FIG. 28 shows a dependence of whether a fixed point iteration willconverge on the value of the first derivative of a function g(x) in theneighborhood of the fixed point.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

This Detailed Description of Preferred Embodiments is organized intoeleven sections. The first section provides an overview of systems andmethods for trading or investing in groups of DBAR contingent claims.The second section describes in detail some of the important features ofsystems and methods for trading or investing in groups of DBARcontingent claims. The third section of this Detailed Description ofPreferred Embodiments provides detailed descriptions of two preferredembodiments of the present invention: investments in a group of DBARcontingent claims, and investments in a portfolio of groups of suchclaims. The fourth section discusses methods for calculating risksattendant on investments in groups and portfolios of groups of DBARcontingent claims. The fifth section of this Detailed Descriptionaddresses liquidity and price/quantity relationships in preferredembodiments of systems and methods of the present invention. The sixthsection provides a detailed description of a DBAR Digital OptionsExchange. The seventh section provides a detailed description of anotherembodiment of a DBAR Digital Options Exchange. The eighth sectionpresents a network implementation of this DBAR Digital Options Exchange.The ninth section presents a structured instrument implementation of ademand-based market or auction. The tenth section presents a detaileddescription of the figures accompanying this specification. The eleventhsection of the Detailed Description discusses some of the salientadvantages of the methods and systems of the present invention. Thetwelfth section is a Technical Appendix providing additional informationon the multistate allocation method of the present invention. The lastsection is a conclusion of the DETAILED DESCRIPTION

More specifically, this Detailed Description of the PreferredEmbodiments is organized as follows:

1 Overview: Exchanges and Markets for DBAR Contingent claims

-   -   1.1 Exchange Design    -   1.2 Market Operation    -   1.3 Network Implementation

2 Features of DBAR Contingent claims

-   -   2.1 DBAR Contingent Claim Notation    -   2.2 Units of Investment and Payouts    -   2.3 Canonical Demand Reallocation Functions    -   2.4 Computing Investment Amounts to Achieve Desired Payouts    -   2.5 A Canonical DRF Example    -   2.6 Interest Considerations    -   2.7 Returns and Probabilities    -   2.8 Computations When Invested Amounts are Large

3 Examples of Groups of DBAR Contingent claims

-   -   3.1 DBAR Range Derivatives    -   3.2 DBAR Portfolios

4 Risk Calculations in Groups of DBAR Contingent claims

-   -   4.1 Market Risk        -   4.1.1 Capital-At-Risk Determinations        -   4.1.2 Capital-At-Risk Determinations Using Monte Carlo            Simulation Techniques        -   4.1.3 Capital-At-Risk Determinations Using Historical            Simulation Techniques    -   4.2 Credit Risk        -   4.2.1 Credit-Capital-At-Risk Determinations        -   4.2.2 Credit-Capital-At-Risk Determinations using Monte            Carlo Simulation Techniques        -   4.2.3 Credit-Capital-At-Risk Historical Simulation            Techniques

5 Liquidity and Price/Quantity Relationships

6 DBAR Digital Options Exchange

-   -   6.1 Representation of Digital Options as DBAR Contingent Claims    -   6.2 Construction of Digital Options Using DBAR Methods and        Systems    -   6.3 Digital Option Spreads    -   6.4 Digital Option Strips    -   6.5 Multistate Allocation Algorithm for Replicating “Sell”        Trades    -   6.6 Clearing and Settlement    -   6.7 Contract Initialization    -   6.8 Conditional Investments, or Limit Orders    -   6.9 Sensitivity Analysis and Depth of Limit Order Book    -   6.10 Networking of DBAR Digital Options Exchanges

7 DBAR DOE: Another Embodiment

-   -   7.1 Special Notation    -   7.2 Elements of Example DBAR DOE Embodiment    -   7.3 Mathematical Principles    -   7.4 Equilibrium Algorithm    -   7.5 Sell Orders    -   7.6 Arbitrary Payout Options    -   7.7 Limit Order Book Optimization    -   7.8 Transaction Fees    -   7.9 An Embodiment of the Algorithm to Solve the Limit Order Book        Optimization    -   7.10 Limit Order Book Display    -   7.11 Unique Price Equilibrium Proof

8 Network Implementation

9 Structured Instrument Trading

-   -   9.1 Overview: Customer Oriented DBAR-enabled Products    -   9.2 Overview: FRNs and swaps    -   9.3 Parameters: FRNs and swaps vs. digital options    -   9.4 Mechanics: DBAR-enabling FRNs and swaps    -   9.5 Example: Mapping FRNs into Digital Option Space    -   9.6 Conclusion

10 Detailed Description of the Drawings

11 Advantages of Preferred Embodiments

12 Technical Appendix

13 Conclusion

In this specification, including the description of preferredembodiments of the present invention, specific terminology will be usedfor the sake of clarity. However, the invention is not intended to belimited to the specific terms so used, and it is to be understood thateach specific term includes all equivalents.

1. OVERVIEW Exchanges and Markets for DBAR Contingent Claims

1.1 Exchange Design

This section describes preferred methods for structuring DBAR contingentclaims and for designing exchanges for the trading of such claims. Thedesign of the exchange is important for effective contingent claimsinvestment in accordance with the present invention. Preferredembodiments of such systems include processes for establishing definedstates and allocating returns, as described below.

-   -   (a) Establishing Defined States: In a preferred embodiment, a        distribution of possible outcomes for an observable event is        partitioned into defined ranges or states. In a preferred        embodiment, one state always occurs because the states are        mutually exclusive and collectively exhaustive. Traders in such        an embodiment invest on their expectation of a return resulting        from the occurrence of a particular outcome within a selected        state. Such investments allow traders to hedge the possible        outcomes of real-world events of economic significance        represented by the states. In a preferred embodiment of a group        of DBAR contingent claims, unsuccessful trades or investments        finance the successful trades or investments. In such an        embodiment the states for a given contingent claim preferably        are defined in such a way that the states are mutually exclusive        and form the basis of a probability distribution, namely, the        sum of the probabilities of all the uncertain outcomes is unity.        For example, states corresponding to stock price closing values        can be established to support a group of DBAR contingent claims        by partitioning the distribution of possible closing values for        the stock on a given future date into ranges. The distribution        of future stock prices, discretized in this way into defined        states, forms a probability distribution in the sense that each        state is mutually exclusive, and the sum of the probabilities of        the stock closing within each defined state at the given date is        unity.        -   In a preferred embodiment, traders can simultaneously invest            in selected multiple states within a given distribution,            without immediately breaking up their investment to fit into            each defined states selected for investment. Traders thus            may place multi-state investments in order to replicate a            desired distribution of returns from a group of contingent            claims. This may be accomplished in a preferred embodiment            of a DBAR exchange through the use of suspense accounts in            which multi-state investments are tracked and reallocated            periodically as returns adjust in response to amounts            invested during a trading period. At the end of a given            trading period, a multi-state investment may be reallocated            to achieve the desired distribution of payouts based upon            the final invested amounts across the distribution of            states. Thus, in such a preferred embodiment, the invested            amount allocated to each of the selected states, and the            corresponding respective returns, are finalized only at the            closing of the trading period. An example of a multi-state            investment illustrating the use of such a suspense account            is provided in Example 3.1.2, below. Other examples of            multi-state investments are provided in Section 6, below,            which describes embodiments of the present invention that            implement DBAR Digital Options Exchanges.    -   (b) Allocating Returns: In a preferred embodiment of a group of        DBAR contingent claims according to the present invention,        returns for each state are specified. In such an embodiment,        while the amount invested for a given trade may be fixed, the        return is adjustable. Determination of the returns for a        particular state can be a simple function of the amount invested        in that state and the total amount invested for all of the        defined states for a group of contingent claims. However,        alternate preferred embodiments can also accommodate methods of        return determination that include other factors in addition to        the invested amounts. For example, in a group of DBAR contingent        claims where unsuccessful investments fund returns to successful        investments, the returns can be allocated based on the relative        amounts invested in each state and also on properties of the        outcome, such as the magnitude of the price changes in        underlying securities. An example in section 3.2 below        illustrates such an embodiment in the context of a securities        portfolio.    -   (c) Determining Investment Amounts: In another embodiment, a        group of DBAR contingent claims can be modeled as digital        options, providing a predetermined or defined payout if they        expire in-the-money, and providing no payout if they expire        out-of-the-money. In this embodiment, the investor or trader        specifies a requested payout for a DBAR digital option, and        selects the outcomes for which the digital option will expire        “in the money,” and can specify a limit on the amount they wish        to invest in such a digital option. Since the payout amount per        digital option (or per an order for a digital option) is        predetermined or defined, investment amounts for each digital        option are determined at the end of the trading period along        with the allocation of payouts per digital option as a function        of the requested payouts, selected outcomes (and limits on        investment amounts, if any) for each of the digital options        ordered during the trading period, and the total amount invested        in the auction or market. This embodiment is described in        Section 7 below, along with another embodiment of demand-based        markets or auctions for digital options described in Section 6        below.

1.2 Market Operation

-   -   (a) Termination Criteria: In a preferred embodiment of a method        of the present invention, returns to investments in the        plurality of defined states are allocated (and in another        embodiment for DBAR digital options, investment amounts are        determined) after the fulfillment of one or more predetermined        termination criteria. In preferred embodiments, these criteria        include the expiration of a “trading period” and the        determination of the outcome of the relevant event after an        “observation period.” In the trading period, traders invest on        their expectation of a return resulting from the occurrence of a        particular outcome within a selected defined state, such as the        state that IBM stock will close between 120 and 125 on. Jul.        6, 1999. In a preferred embodiment, the duration of the trading        period is known to all participants; returns associated with        each state vary during the trading period with changes in        invested amounts; and returns are allocated based on the total        amount invested in all states relative to the amounts invested        in each of the states as at the end of the trading period.        -   Alternatively, the duration of the trading period can be            unknown to the participants. The trading period can end, for            example, at a randomly selected time. Additionally, the            trading period could end depending upon the occurrence of            some event associated or related to the event of economic            significance, or upon the fulfillment of some criterion. For            example, for DBAR contingent claims traded on reinsurance            risk (discussed in Section 3 below), the trading period            could close after an nth catastrophic natural event (e.g., a            fourth hurricane), or after a catastrophic event of a            certain magnitude (e.g., an earthquake of a magnitude of 5.5            or higher on the Richter scale). The trading period could            also close after a certain volume, amount, or frequency of            trading is reached in a respective auction or market.        -   The observation period can be provided as a time period            during which the contingent events are observed and the            relevant outcomes determined for the purpose of allocating            returns. In a preferred embodiment, no trading occurs during            the observation period.        -   The expiration date, or “expiration,” of a group of DBAR            contingent claims as used in this specification occurs when            the termination criteria are fulfilled for that group of            DBAR contingent claims. In a preferred embodiment, the            expiration is the date, on or after the occurrence of the            relevant event, when the outcome is ascertained or observed.            This expiration is similar to well-known expiration features            in traditional options or futures in which a future date,            i.e., the expiration date, is specified as the date upon            which the value of the option or future will be determined            by reference to the value of the underlying financial            product on the expiration date.        -   The duration of a contingent claim as defined for purposes            of this specification is simply the amount of time remaining            until expiration from any given reference date. A trading            start date (“TSD”) and a trading end date (“TED”), as used            in the specification, refer to the beginning and end of a            time period (“trading period”) during which traders can make            investments in a group of DBAR contingent claims. Thus, the            time during which a group of DBAR contingent claims is open            for investment or trading, i.e., the difference between the            TSD and TED, may be referred to as the trading period. In            preferred embodiments, there can be one or many trading            periods for a given expiration date, opening successively            through time. For example, one trading period's TED may            coincide exactly with the subsequent trading period's TSD,            or in other examples, trading periods may overlap.        -   The relationship between the duration of a contingent claim,            the number of trading periods employed for a given event,            and the length and timing of the trading periods, can be            arranged in a variety of ways to maximize trading or achieve            other goals. In preferred embodiments at least one trading            period occurs—that is, starts and ends—prior in time to the            identification of the outcome of the relevant event. In            other words, in preferred embodiments, the trading period            will most likely temporally precede the event defining the            claim. This need not always be so, since the outcome of an            event may not be known for some time thereby enabling            trading periods to end (or even start) subsequent to the            occurrence of the event, but before its outcome is known.        -   A nearly continuous or “quasi-continuous” market can be made            available by creating multiple trading periods for the same            event, each having its own closing returns. Traders can make            investments during successive trading periods as the returns            change. In this way, profits-and-losses can be realized at            least as frequently as in current derivatives markets. This            is how derivatives traders currently are able to hedge            options, futures, and other derivatives trades. In preferred            embodiments of the present invention, traders may be able to            realize profits and at varying frequencies, including more            frequently than daily.    -   (b) Market Efficiency and Fairness: Market prices reflect, among        other things, the distribution of information available to        segments of the participants transacting in the market. In most        markets, some participants will be better informed than others.        In house-banking or traditional markets, market makers protect        themselves from more informed counterparties by increasing their        bid-offer spreads.        -   In preferred embodiments of DBAR contingent claim markets,            there may be no market makers as such who need to protect            themselves. It may nevertheless be necessary to put in place            methods of operation in such markets in order to prevent            manipulation of the outcomes underlying groups of DBAR            contingent claims or the returns payable for various            outcomes. One such mechanism is to introduce an element of            randomness as to the time at which a trading period closes.            Another mechanism to minimize the likelihood and effects of            market manipulation is to introduce an element of randomness            to the duration of the observation period. For example, a            DBAR contingent claim might settle against an average of            market closing prices during a time interval that is            partially randomly determined, as opposed to a market            closing price on a specific day.        -   Additionally, in preferred embodiments incentives can be            employed in order to induce traders to invest earlier in a            trading period rather than later. For example, a DRF may be            used which allocates slightly higher returns to earlier            investments in a successful state than later investments in            that state. For DBAR digital options, an OPF may be used            which determines slightly lower (discounted) prices for            earlier investments than later investments. Earlier            investments may be valuable in preferred embodiments since            they work to enhance liquidity and promote more uniformly            meaningful price information during the trading period.    -   (c) Credit Risk: In preferred embodiments of a DBAR contingent        claims market, the dealer or exchange is substantially protected        from primary market risk by the fundamental principle underlying        the operation of the system—that returns to successful        investments are funded by losses from unsuccessful investments.        The credit risk in such preferred embodiments is distributed        among all the market participants. If, for example, leveraged        investments are permitted within a group of DBAR contingent        claims, it may not be possible to collect the leveraged        unsuccessful investments in order to distribute these amounts        among the successful investments.        -   In almost all such cases there exists, for any given trader            within a group of DBAR contingent claims, a non-zero            possibility of default, or credit risk. Such credit risk is,            of course, ubiquitous to all financial transactions            facilitated with credit.        -   One way to address this risk is to not allow leveraged            investments within the group of DBAR contingent claims,            which is a preferred embodiment of the system and methods of            the present invention. In other preferred embodiments,            traders in a DBAR exchange may be allowed to use limited            leverage, subject to real-time margin monitoring, including            calculation of a trader's impact on the overall level of            credit risk in the DBAR system and the particular group of            contingent claims. These risk management calculations should            be significantly more tractable and transparent than the            types of analyses credit risk managers typically perform in            conventional derivatives markets in order to monitor            counterparty credit risk.        -   An important feature of preferred embodiments of the present            invention is the ability to provide diversification of            credit risk among all the traders who invest in a group of            DBAR contingent claims. In such embodiments, traders make            investments (in the units of value as defined for the group)            in a common distribution of states in the expectation of            receiving a return if a given state is determined to have            occurred. In preferred embodiments, all traders, through            their investments in defined states for a group of            contingent claims, place these invested amounts with a            central exchange or intermediary which, for each trading            period, pays the returns to successful investments from the            losses on unsuccessful investments. In such embodiments, a            given trader has all the other traders in the exchange as            counterparties, effecting a mutualization of counterparties            and counterparty credit risk exposure. Each trader therefore            assumes credit risk to a portfolio of counterparties rather            than to a single counterparty.        -   Preferred embodiments of the DBAR contingent claim and            exchange of the present invention present four principal            advantages in managing the credit risk inherent in leveraged            transactions. First, a preferred form of DBAR contingent            claim entails limited liability investing. Investment            liability is limited in these embodiments in the sense that            the maximum amount a trader can lose is the amount invested.            In this respect, the limited liability feature is similar to            that of a long option position in the traditional markets.            By contrast, a short option position in traditional markets            represents a potentially unlimited liability investment            since the downside exposure can readily exceed the option            premium and is, in theory, unbounded. Importantly, a group            of DBAR contingent claims of the present invention can            easily replicate returns of a traditional short option            position while maintaining limited liability. The limited            liability feature of a group of DBAR contingent claims is a            direct consequence of the demand-side nature of the market.            More specifically, in preferred embodiments there are no            sales or short positions as there are in the traditional            markets, even though traders in a group of DBAR contingent            claims may be able to attain the return profiles of            traditional short positions.        -   Second, in preferred embodiments, a trader within a group of            DBAR contingent claims should have a portfolio of            counterparties as described above. As a consequence, there            should be a statistical diversification of the credit risk            such that the amount of credit risk borne by any one trader            is, on average (and in all but exceptionally rare cases),            less than if there were an exposure to a single counterparty            as is frequently the case in traditional markets. In other            words, in preferred embodiments of the system and methods of            the present invention, each trader is able to take advantage            of the diversification effect that is well known in            portfolio analysis.        -   Third, in preferred embodiments of the present invention,            the entire distribution of margin loans, and the aggregate            amount of leverage and credit risk existing for a group of            DBAR contingent claims, can be readily calculated and            displayed to traders at any time before the fulfillment of            all of the termination criteria for the group of claims.            Thus, traders themselves may have access to important            information regarding credit risk. In traditional markets            such information is not readily available.        -   Fourth, preferred embodiments of a DBAR contingent claim            exchange provide more information about the distribution of            possible outcomes than do traditional market exchanges.            Thus, as a byproduct of DBAR contingent claim trading            according to preferred embodiments, traders have more            information about the distribution of future possible            outcomes for real-world events, which they can use to manage            risk more effectively. For many traders, a significant part            of credit risk is likely to be caused by market risk. Thus,            in preferred embodiments of the present invention, the            ability through an exchange or otherwise to control or at            least provide information about market risk should have            positive feedback effects for the management of credit risk.

A simple example of a group of DBAR contingent claims with the followingassumptions, illustrates some of these features. The example uses thefollowing basic assumptions:

-   -   two defined states (with predetermined termination        criteria): (i) stock price appreciates in one month; (ii) stock        price depreciates in one month; and    -   $100 has been invested in the appreciate state, and $95 in the        depreciate state.

If a trader then invests $1 in the appreciate state, if the stock infact appreciates in the month, then the trader will be allocated apayout of $1.9406 (= 196/101)—a return of $0.9406 plus the original $1investment (ignoring, for the purpose of simplicity in thisillustration, a transaction fee). If, before the close of the tradingperiod the trader desires effectively to “sell” his investment in theappreciate state, he has two choices. He could sell the investment to athird party, which would necessitate crossing of a bid and an offer in atwo-way order crossing network. Or, in a preferred embodiment of themethod of the present invention, the trader can invest in the depreciatestate, in proportion to the amount that had been invested in that statenot counting the trader's “new” investments. In this example, in orderto fully hedge his investment in the appreciate state, the trader caninvest $0.95 ( 95/100) in the depreciate state. Under either possibleoutcome, therefore, the trader will receive a payout of $1.95, i.e., ifthe stock appreciates the trader will receive 196.95/101=$1.95 and ifthe stock depreciates the trader will receive (196.95/95.95)*0.95=$1.95.

1.3 Network Implementation

A market or exchange for groups of DBAR contingent claims marketaccording to the invention is not designed to establish acounterparty-driven or order-matched market. Buyers' bids and sellers'offers do not need to be “crossed.” As a consequence of the absence of aneed for an order crossing network, preferred embodiments of the presentinvention are particularly amenable to large-scale electronic networkimplementation on a wide area network or a private network (with, e.g.,dedicated circuits) or the public Internet, for example.

Preferred embodiments of an electronic network-based embodiment of themethod of trading in accordance with the invention include one or moreof the following features.

-   -   (a) User Accounts: DBAR contingent claims investment accounts        are established using electronic methods.    -   (b) Interest and Margin Accounts: Trader accounts are maintained        using electronic methods to record interest paid to traders on        open DBAR contingent claim balances and to debit trader balances        for margin loan interest. Interest is typically paid on        outstanding investment balances for a group of DBAR contingent        claims until the fulfillment of the termination criteria.        Interest is typically charged on outstanding margin loans while        such loans are outstanding. For some contingent claims, trade        balance interest can be imputed into the closing returns of a        trading period.    -   (c) Suspense Accounts: These accounts relate specifically to        investments which have been made by traders, during trading        periods, simultaneously in multiple states for the same event.        Multi-state trades are those in which amounts are invested over        a range of states so that, if any of the states occurs, a return        is allocated to the trader based on the closing return for the        state which in fact occurred. DBAR digital options of the        present invention, described in Section 6, provide other        examples of multi-state trades.        -   A trader can, of course, simply break-up or divide the            multi-state investment into many separate, single-state            investments, although this approach might require the trader            to keep rebalancing his portfolio of single state            investments as returns adjust throughout the trading period            as amounts invested in each state change.        -   Multi-state trades can be used in order to replicate any            arbitrary distribution of payouts that a trader may desire.            For example, a trader might want to invest in all states in            excess of a given value or price for a security underlying a            contingent claim, e.g., the occurrence that a given stock            price exceeds 100 at some future date. The trader might also            want to receive an identical payout no matter what state            occurs among those states. For a group of DBAR contingent            claims there may well be many states for outcomes in which            the stock price exceeds 100 (e.g., greater than 100 and less            than or equal to 101; greater than 101 and less than or            equal to 102, etc.). In order to replicate a multi-state            investment using single state investments, a trader would            need continually to rebalance the portfolio of single-state            investments so that the amount invested in the selected            multi-states is divided among the states in proportion to            the existing amount invested in those states. Suspense            accounts can be employed so that the exchange, rather than            the trader, is responsible for rebalancing the portfolio of            single-state investments so that, at the end of the trading            period, the amount of the multi-state investment is            allocated among the constituent states in such a way so as            to replicate the trader's desired distribution of payouts.            Example 3.1.2 below illustrates the use of suspense accounts            for multi-state investments.    -   (d) Authentication: Each trader may have an account that may be        authenticated using authenticating data.    -   (e) Data Security: The security of contingent claims        transactions over the network may be ensured, using for example        strong forms of public and private key encryption.    -   (f) Real-Time Market Data Server: Real-time market data may be        provided to support frequent calculation of returns and to        ascertain the outcomes during the observation periods.    -   (g) Real-Time Calculation Engine Server: Frequent calculation of        market returns may increase the efficient functioning of the        market. Data on coupons, dividends, market interest rates, spot        prices, and other market data can be used to calculate opening        returns at the beginning of a trading period and to ascertain        observable events during the observation period. Sophisticated        simulation methods may be required for some groups of DBAR        contingent claims in order to estimate expected returns, at        least at the start of a trading period.    -   (h) Real-Time Risk Management Server: In order to compute trader        margin requirements, expected returns for each trader should be        computed frequently. Calculations of “value-at-risk” in        traditional markets can involve onerous matrix calculations and        Monte Carlo simulations. Risk calculations in preferred        embodiments of the present invention are simpler, due to the        existence of information on the expected returns for each state.        Such information is typically unavailable in traditional capital        and reinsurance markets.    -   (i) Market Data Storage: A DBAR contingent claims exchange in        accordance with the invention may generate valuable data as a        byproduct of its operation. These data are not readily available        in traditional capital or insurance markets. In a preferred        embodiment of the present invention, investments may be        solicited over ranges of outcomes for market events, such as the        event that the 30-year U.S. Treasury bond will close on a given        date with a yield between 6.10% and 6.20%. Investment in the        entire distribution of states generates data that reflect the        expectations of traders over the entire distribution pf possible        outcomes. The network implementation disclosed in this        specification may be used to capture, store and retrieve these        data.    -   (j) Market Evaluation Server: Preferred embodiments of the        method of the present invention include the ability to improve        the market's efficiency on an ongoing basis. This may readily be        accomplished, for example, by comparing the predicted returns on        a group of DBAR contingent claims returns with actual realized        outcomes. If investors have rational expectations, then DBAR        contingent claim returns will, on average, reflect trader        expectations, and these expectations will themselves be realized        on average. In preferred embodiments, efficiency measurements        are made on defined states and investments over the entire        distribution of possible outcomes, which can then be used for        statistical time series analysis with realized outcomes. The        network implementation of the present invention may therefore        include analytic servers to perform these analyses for the        purpose of continually improving the efficiency of the market.

2. Features of DBAR Contingent Claims

In a preferred embodiment, a group of a DBAR contingent claims relatedto an observable event includes one or more of the following features:

-   -   (1) A defined set of collectively exhaustive states representing        possible real-world outcomes related to an observable event. In        preferred embodiments, the events are events of economic        significance. The possible outcomes can typically be units of        measurement associated with the event, e.g., an event of        economic interest can be the closing index level of the S&P 500        one month in the future, and the possible outcomes can be entire        range of index levels that are possible in one month. In a        preferred embodiment, the states are defined to correspond to        one or more of the possible outcomes over the entire range of        possible outcomes, so that defined states for an event form a        countable and discrete number of ranges of possible outcomes,        and are collectively exhaustive in the sense of spanning the        entire range of possible outcomes. For example, in a preferred        embodiment, possible outcomes for the S&P 500 can range from        greater than 0 to infinity (theoretically), and a defined state        could be those index values greater than 1000 and less than or        equal to 1100. In such preferred embodiments, exactly one state        occurs when the outcome of the relevant event becomes known.    -   (2) The ability for traders to place trades on the designated        states during one or more trading periods for each event. In a        preferred embodiment, a DBAR contingent claim group defines the        acceptable units of trade or value for the respective claim.        Such units may be dollars, barrels of oil, number of shares of        stock, or any other unit or combination of units accepted by        traders and the exchange for value.    -   (3) An accepted determination of the outcome of the event for        determining which state or states have occurred. In a preferred        embodiment, a group of DBAR contingent claims defines the means        by which the outcome of the relevant events is determined. For        example, the level that the S&P 500 Index actually closed on a        predetermined date would be an outcome observation which would        enable the determination of the occurrence of one of the defined        states. A closing value of 1050 on that date, for instance,        would allow the determination that the state between 1000 and        1100 occurred.    -   (4) The specification of a DRF which takes the traded amount for        each trader for each state across the distribution of states as        that distribution exists at the end of each trading period and        calculates payouts for each investments in each state        conditioned upon the occurrence of each state. In preferred        embodiments, this is done so that the total amount of payouts        does not exceed the total amount invested by all the traders in        all the states. The DRF can be used to show payouts should each        state occur during the trading period, thereby providing to        traders information as to the collective level of interest of        all traders in each state.    -   (5) For DBAR digital options, the specification of an OPF which        takes the requested payout and selection of outcomes and limits        on investment amounts (if any) per digital option at the end of        each trading period and calculates the investment amounts per        digital option, along with the payouts for each digital option        in each state conditioned upon the occurrence of each state. In        this other embodiment, this is done by solving a nonlinear        optimization problem which uses the DRF along with a series of        other parameters to determine an optimal investment amount per        digital option while maximizing the possible payout per digital        option.    -   (6) Payouts to traders for successful investments based on the        total amount of the unsuccessful investments after deduction of        the transaction fee and after fulfillment of the termination        criteria.    -   (7) For DBAR digital options, investment amounts per digital        option after factoring in the transaction fee and after        fulfillment of the termination criteria.

The states corresponding to the range of possible event outcomes arereferred to as the “distribution” or “distribution of states.” Each DBARcontingent claim group or “contract” is typically associated with onedistribution of states. The distribution will typically be defined forevents of economic interest for investment by traders having theexpectation of a return for a reduction of risk (“hedging”), or for anincrease of risk (“speculation”). For example, the distribution can bebased upon the values of stocks, bonds, futures, and foreign exchangerates. It can also be based upon the values of commodity indices,economic statistics (e.g., consumer price inflation monthly reports),property-casualty losses, weather patterns for a certain geographicalregion, and any other measurable or observable occurrence or any otherevent in which traders would not be economically indifferent even in theabsence of a trade on the outcome of the event.

2.1 DBAR Claim Notation

The following notation is used in this specification to facilitatefurther description of DBAR contingent claims:

-   -   m represents the number of traders for a given group of DBAR        contingent    -   n represents the number of states for a given distribution        associated with a given group of DBAR contingent claims    -   A represents a matrix with m rows and n columns, where the        element at the i-th row and j-th column, α_(i,j), is the amount        that trader i has invested in state j in the expectation of a        return should state j occur    -   Π represents a matrix with n rows and n columns where element        π_(i,j) is the payout per unit of investment in, state i should        state j occur (“unit payouts”)    -   R represents a matrix with n rows and n columns where element        r_(i,j) is the return per unit of investment in state i should        state j occur, i.e., r_(i,j)=π_(i,j)−1 (“unit returns”)    -   P represents a matrix with m rows and n columns, where the        element at the i-th row and j-th column, p_(i,j), is the payout        to be made to trader i should state j occur, i.e., P is equal to        the matrix product A*Π.    -   P_(*j,) represents the j-th column of P, for j=1 . . . n, which        contains the payouts to each investment should state j occur    -   P_(i,*) represents the i-th row of P, for i=1 . . . m, which        contains the payouts to trader i    -   s_(i) where i=1 . . . n, represents a state representing a range        of possible outcomes of an observable event.    -   T_(i) where i=1 . . . n, represents the total amount traded in        the expectation of the occurrence of state i    -   T represents the total traded amount over the entire        distribution of states, i.e.,

$T = {\sum\limits_{i = {1\mspace{14mu} \ldots \mspace{14mu} n}}T_{i}}$

-   -   f(A,X) represents the exchange's transaction fee, which can        depend on the entire distribution of traded amounts placed        across all the states as well as other factors, X, some of which        are identified below. For reasons of brevity, for the remainder        of this specification unless otherwise stated, the transaction        fee is assumed to be a fixed percentage of the total amount        traded over all the states.    -   c_(p) represents the interest rate charged on margin loans.    -   c_(r) represents the interest rate paid on trade balances.    -   t represents time from the acceptance of a trade or investment        to the fulfillment of all of the termination criteria for the        group of DBAR contingent claims, typically expressed in years or        fractions thereof.    -   X represents other information upon which the DRF or transaction        fee can depend such as information specific to an investment or        a trader, including for example the time or size of a trade.

In preferred embodiments, a DRF is a function that takes the tradedamounts over the distribution of states for a given group of DBARcontingent claims, the transaction fee schedule, and, conditional uponthe occurrence of each state, computes the payouts to each trade orinvestment placed over the distribution of states. In notation, such aDRF is:

P=DRF(A,f(A,X),X|s=s _(i))=A*Π(A,f(A,X),X)  (DRF)

In other words, the m traders who have placed trades across the nstates, as represented in matrix A, will receive payouts as representedin matrix P should state i occur, also, taking into account thetransaction fee f and other factors X. The payouts identified in matrixP can be represented as the product of (a) the payouts per unit tradedfor each state should each state occur, as identified in the matrix Π,and (b) the matrix A which identifies the amounts traded or invested byeach trader in each state. The following notation may be used toindicate that, in preferred embodiments, payouts should not exceed thetotal amounts invested less the transaction fee, irrespective of whichstate occurs:

1_(m) ^(T) *P _(*,j) +f(A,X)<=1_(m) ^(T) *A*1_(n) for j=1 . . . n  (DRFConstraint)

where the 1 represents a column vector with dimension indicated by thesubscript, the superscript T represents the standard transpose operatorand P_(*,j) is the j-th column of the matrix P representing the payoutsto be made to each trader should state j occur. Thus, in preferredembodiments, the unsuccessful investments finance the successfulinvestments. In addition, absent credit-related risks discussed below,in such embodiments there is no risk that payouts will exceed the totalamount invested in the distribution of states, no matter what stateoccurs. In short, a preferred embodiment of a group of DBAR contingentclaims of the present invention is self-financing in the sense that forany state, the payouts plus the transaction fee do not exceed the inputs(i.e., the invested amounts).

The DRF may depend on factors other than the amount of the investmentand the state in which the investment was made. For example, a payoutmay depend upon the magnitude of a change in the observed outcome for anunderlying event between two dates (e.g., the change in price of asecurity between two dates). As another example, the DRF may allocatehigher payouts to traders who initiated investments earlier in thetrading period than traders who invested later in the trading period,thereby providing incentives for liquidity earlier in the tradingperiod. Alternatively, the DRF may allocate higher payouts to largeramounts invested in a given state than to smaller amounts invested forthat state, thereby providing another liquidity incentive.

In any event, there are many possible functional forms for a DRF thatcould be used. To illustrate, one trivial form of a DRF is the case inwhich the traded amounts, A, are not reallocated at all upon theoccurrence of any state, i.e., each trader receives his traded amountback in the event that any state occurs, as indicated by the followingnotation:

P=A if s=s _(i), for i=1 . . . n

This trivial DRF is not useful in allocating and exchanging risk amonghedgers.

For a meaningful risk exchange to occur, a preferred embodiment of a DRFshould effect a meaningful reallocation of amounts invested across thedistribution of states upon the occurrence of at least one state. Groupsof DBAR contingent claims of the present invention are discussed in thecontext of a canonical DRF, which is a preferred embodiment in which theamounts invested in states which did not occur are completelyreallocated to the state which did occur (less any transaction fee). Thepresent invention is not limited to a canonical DRF, and many othertypes of DRFs can be used and may be preferred to implement a group ofDBAR contingent claims. For example, another DRF preferred embodimentallocates half the total amount invested to the outcome state andrebates the remainder of the total amount invested to the states whichdid not occur. In another preferred embodiment, a DRF would allocatesome percentage to an occurring state, and some other percentage to oneor more “nearby” or “adjacent” states with the bulk of the non-occurringstates receiving zero payouts. Section 7 decribes an OPF for DBARdigital options which includes a DRF and determines investment amountsper investment or order along with allocating returns. Other DRFs willbe apparent to those of skill in the art from review of thisspecification and practice of the present invention.

2.2 Units of Investments and Payouts

The units of investments and payouts in systems and methods of thepresent invention may be units of currency, quantities of commodities,numbers of shares of common stock, amount of a swap transaction or anyother units representing economic value. Thus, there is no limitationthat the investments or payouts be in units of currency or money (e.g.,U.S. dollars) or that the payouts resulting from the DRF be in the sameunits as the investments. Preferably, the same unit of value is used torepresent the value of each investment, the total amount of allinvestments in a group of DBAR contingent claims, and the amountsinvested in each state.

It is possible, for example, for traders to make investments in a groupof DBAR contingent claims in numbers of shares of common stock and forthe applicable DRF (or OPF) to allocate payouts to traders in JapaneseYen or barrels of oil. Furthermore, it is possible for traded amountsand payouts to be some combination of units, such as, for example, acombination of commodities, currencies, and number of shares. Inpreferred embodiments traders need not physically deposit or receivedelivery of the value units, and can rely upon the DBAR contingent claimexchange to convert between units for the purposes of facilitatingefficient trading and payout transactions. For example, a DBARcontingent claim might be defined in such a way so that investments andpayouts are to be made in ounces of gold. A trader can still depositcurrency, e.g., U.S. dollars, with the exchange and the exchange can beresponsible for converting the amount invested in dollars into thecorrect units, e.g., gold, for the purposes of investing in a givenstate or receiving a payout. In this specification, a U.S. dollar istypically used as the unit of value for investments and payouts. Thisinvention is not limited to investments or payouts in that value unit.In situations where investments and payouts are made in different unitsor combinations of units, for purpose of allocating returns to eachinvestment the exchange preferably converts the amount of eachinvestment, and thus the total of the investments in a group of DBARcontingent claims, into a single unit of value (e.g., dollars). Example3.1.20 below illustrates a group of DBAR contingent claims in whichinvestments and payouts are in units of quantities of common stockshares.

2.3 Canonical Demand Reallocation Function

A preferred embodiment of a DRF that can be used to implement a group ofDBAR contingent claims is termed a “canonical” DRF. A canonical DRF is atype of DRF which has the following property: upon the occurrence of agiven state I, investors who have invested in that state receive apayout per unit invested equal to (a) the total amount traded for allthe states less the transaction fee, divided by (b) the total amountinvested in the occurring state. A canonical DRF may employ atransaction fee which may be a fixed percentage of the total amounttraded, T, although other transaction fees are possible. Traders whomade investments in states which not did occur receive zero payout.Using the notation developed above:

$\pi_{i,j} = \frac{\left( {1 - f} \right)*T}{T_{i}}$

if i=j, i.e., the unit payout to an investment in state i if state ioccurs

-   π_(i,j)=0 otherwise, i.e., if i≠j, so that the payout is zero to    investments in state i if state j occurs.    In a preferred embodiment of a canonical DRF, the unit payout matrix    Π as defined above is therefore a diagonal matrix with entries equal    to π_(i,j) for i=j along the diagonal, and zeroes for all    off-diagonal entries. For example, in a preferred embodiment for n=5    states, the unit payout matrix is:

$\begin{matrix}{\Pi = {\begin{bmatrix}\frac{T}{T_{1}} & 0 & 0 & 0 & 0 \\0 & \frac{T}{T_{2}} & 0 & 0 & 0 \\0 & 0 & \frac{T}{T_{3}} & 0 & 0 \\0 & 0 & 0 & \frac{T}{T_{4}} & 0 \\0 & 0 & 0 & 0 & \frac{T}{T_{5}}\end{bmatrix}*\left( {1 - f} \right)}} \\{= {\begin{bmatrix}\frac{1}{T_{1}} & 0 & 0 & 0 & 0 \\0 & \frac{1}{T_{2}} & 0 & 0 & 0 \\0 & 0 & \frac{1}{T_{3}} & 0 & 0 \\0 & 0 & 0 & \frac{1}{T_{4}} & 0 \\0 & 0 & 0 & 0 & \frac{1}{T_{5}}\end{bmatrix}*T*\left( {1 - f} \right)}}\end{matrix}$

For this embodiment of a canonical DRF, the payout matrix is the totalamount invested less the transaction fee, multiplied by a diagonalmatrix which contains the inverse of the total amount invested in eachstate along the diagonal, respectively, and zeroes elsewhere. Both T,the total amount invested by all m traders across all n states, andT_(i), the total amount invested in state I, are functions of the matrixA, which contains the amount each trader has invested in each state:

T _(i)=1_(m) ^(T) *A*B _(n)(i)

T=1_(m) ^(T) *A*1_(n)

where B_(n)(i) is a column vector of dimension n which has a 1 at thei-th row and zeroes elsewhere. Thus, with n=5 as an example, thecanonical DRF described above has a unit payout matrix which is afunction of the amounts traded across the states and the transactionfee:

$\Pi = {\begin{bmatrix}\frac{1}{1_{m}^{T}*A*{B_{n}(1)}} & 0 & 0 & 0 & 0 \\0 & \frac{1}{1_{m}^{T}*A*{B_{n}(2)}} & 0 & 0 & 0 \\0 & 0 & \frac{1}{1_{m}^{T}*A*{B_{n}(3)}} & 0 & 0 \\0 & 0 & 0 & \frac{1}{1_{m}^{T}*A*{B_{n}(4)}} & 0 \\0 & 0 & 0 & 0 & \frac{1}{1_{m}^{T}*A*{B_{n}(5)}}\end{bmatrix}*1_{m}^{T}*A*1_{n}*\left( {1 - f} \right)}$

which can be generalized for any arbitrary number of states. The actualpayout matrix, in the defined units of value for the group of DBARcontingent claims (e.g., dollars), is the product of the m×n tradedamount matrix A and the n×n unit payout matrix Π, as defined above:

P=A*Π(A,f)  (CDRF)

This provides that the payout matrix as defined above is the matrixproduct of the amounts traded as contained in the matrix A and the unitpayout matrix Π, which is itself a function of the matrix A and thetransaction fee, f. The expression is labeled CDRF for “Canonical DemandReallocation Function.”

It should be noted that, in this preferred embodiment, any change to thematrix A will generally have an effect on any given trader's payout,both due to changes in the amount invested, i.e., a direct effectthrough the matrix A in the CDRF, and changes in the unit payouts, i.e.,an indirect effect since the unit payout matrix Π is itself a functionof the traded amount matrix A.

2.4 Computing Investment Amounts to Achieve Desired Payouts

In preferred embodiments of a group of DBAR contingent claims of thepresent invention, some traders make investments in states during thetrading period in the expectation of a payout upon the occurrence of agiven state, as expressed in the CDRF above. Alternatively, a trader mayhave a preference for a desired payout distribution should a given stateoccur. DBAR digital options, described in Section 6, are an example ofan investment with a desired payout distribution should one or morespecified states occur. Such a payout distribution could be denotedP_(i,*), which is a row corresponding to trader i in payout matrix P.Such a trader may want to know how much to invest in contingent claimscorresponding to a given state or states in order to achieve this payoutdistribution. In a preferred embodiment, the amount or amounts to beinvested across the distribution of states for the CDRF, given a payoutdistribution, can be obtained by inverting the expression for the CDRFand solving for the traded amount matrix A:

A=P*Π(A,f)⁻¹  (CDRF 2)

In this notation, the −1 superscript on the unit payout matrix denotes amatrix inverse.

Expression CDRF 2 does not provide an explicit solution for the tradedamount matrix A, since the unit payout matrix Π is itself a function ofthe traded amount matrix. CDRF 2 typically involves the use of numericalmethods to solve m simultaneous quadratic equations. For example,consider a trader who would like to know what amount, α, should betraded for a given state i in order to achieve a desired payout of p.Using the “forward” expression to compute payouts from traded amounts asin CDRF above yields the following equation:

$p = {\left( \frac{T + \alpha}{T_{i} + \alpha} \right)*\alpha}$

This represents a given row and column of the matrix equation CDRF aftera has been traded for state i (assuming no transaction fee). Thisexpression is quadratic in the traded amount α, and can be solved forthe positive quadratic root as follows:

$\begin{matrix}{\alpha = \frac{\left( {p - T} \right) + \sqrt{\left( {p - T} \right)^{2} + {4*p*T_{i}}}}{2}} & \left( {{CDRF}\mspace{14mu} 3} \right)\end{matrix}$

2.5 A Canonical DRF Example

A simplified example illustrates the use of the CDRF with a group ofDBAR contingent claims defined over two states (e.g., states “1” and“2”) in which four traders make investments. For the example, thefollowing assumptions are made: (1) the transaction fee, f, is zero; (2)the investment and payout units are both dollars; (3) trader 1 has madeinvestments in the amount of $5 in state 1 and $10 state 2; and (4)trader 2 has made an investment in the amount of $7 for state 1 only.With the investment activity so far described, the traded amount matrixA, which as 4 rows and 2 columns, and the unit payout matrix Π which has2 rows and 2 columns, would be denoted as follows:

$A = \begin{matrix}5 & 10 \\7 & 0 \\0 & 0 \\0 & 0\end{matrix}$ $\Pi = {\begin{bmatrix}\frac{1}{12} & 0 \\0 & \frac{1}{10}\end{bmatrix}*22}$

The payout matrix P, which contains the payouts in dollars for eachtrader should each state occur is, the product of A and Π:

$P = \begin{matrix}9.167 & 22 \\12.833 & 0 \\0 & 0 \\0 & 0\end{matrix}$

The first row of P corresponds to payouts to trader 1 based on hisinvestments and the unit payout matrix. Should state 1 occur, trader 1will receive a payout of $9.167 and will receive $22 should state 2occur. Similarly, trader 2 will receive $12.833 should state 1 occur and$0 should state 2 occur (since trader 2 did not make any investment instate 2). In this illustration, traders 3 and 4 have $0 payouts sincethey have made no investments.

In accordance with the expression above labeled “DRF Constraint,” thetotal payouts to be made upon the occurrence of either state is lessthan or equal to the total amounts invested. In other words, the CDRF inthis example is self-financing so that total payouts plus thetransaction fee (assumed to be zero in this example) do not exceed thetotal amounts invested, irrespective of which state occurs. This isindicated by the following notation:

1_(m) ^(T) *P _(*,1)=22≦1_(m) ^(T) *A*1_(n)=22

1_(m) ^(T) *P _(*,2)=22≦1_(m) ^(T) *A*1_(n)=22

Continuing with this example, it is now assumed that traders 3 and 4each would like to make investments that generate a desired payoutdistribution. For example, it is assumed that trader 3 would like toreceive a payout of $2 should state 1 occur and $4 should state 2 occur,while trader 4 would like to receive a payout of $5 should state 1 occurand $0 should state 2 occur. In the CDRF notation:

P _(3,*)=[2 4]

P _(4,*)=[5 0]

In a preferred embodiment and this example, payouts are made based uponthe invested amounts A, and therefore are also based on the unit payoutmatrix Π(A,f(A)), given the distribution of traded amounts as they existat the end of the trading period. For purposes of this example, it isnow further assumed (a) that at the end of the trading period traders 1and 2 have made investments as indicated above, and (b) that the desiredpayout distributions for traders 3 and 4 have been recorded in asuspense account which is used to determine the allocation ofmulti-state investments to each state in order to achieve the desiredpayout distributions for each trader, given the investments by the othertraders as they exist at the end of the trading period. In order todetermine the proper allocation, the suspense account can be used tosolve CDRF 2, for example:

$\begin{bmatrix}5 & 10 \\7 & 0 \\\alpha_{3,1} & \alpha_{3,2} \\\alpha_{4,1} & \alpha_{4,2}\end{bmatrix} = {\begin{bmatrix}p_{1,1} & p_{1,2} \\p_{2,1} & p_{2,2} \\2 & 4 \\5 & 0\end{bmatrix}*{\quad{\left\lbrack \begin{matrix}\frac{1}{\left( {5 + 7 + \alpha_{3,1} + \alpha_{4,1}} \right)} & 0 \\0 & \frac{1}{\left( {10 + 0 + \alpha_{3,2} + \alpha_{4,2}} \right)}\end{matrix} \right\rbrack \frac{{{cont}'}d}{below}*\left( {5 + 10 + 7 + 0 + \alpha_{3,1} + \alpha_{4,1} + \alpha_{3,2} + \alpha_{4,2}} \right)}}}$

The solution of this expression will yield the amounts that traders 3and 4 need to invest in for contingent claims corresponding to states 1and 2 to in order to achieve their desired payout distributions,respectively. This solution will also finalize the total investmentamount so that traders 1 and 2 will be able to determine their payoutsshould either state occur. This solution can be achieved using acomputer program that computes an investment amount for each state foreach trader in order to generate the desired payout for that trader forthat state. In a preferred embodiment, the computer program repeats theprocess iteratively until the calculated investment amounts converge,i.e., so that the amounts to be invested by traders 3 and 4 no longermaterially change with each successive iteration of the computationalprocess. This method is known in the art as fixed point iteration and isexplained in more detail in the Technical Appendix. The following tablecontains a computer code listing of two functions written in Microsoft'sVisual Basic which can be used to perform the iterative calculations tocompute the final allocations of the invested amounts in this example ofa group of DBAR contingent claims with a Canonical Demand ReallocationFunction:

TABLE 1 Illustrative Visual Basic Computer Code for Solving CDRF 2Function allocatetrades(A_mat, P_mat) As Variant Dim A_final Dim tradesAs Long Dim states As Long trades = P_mat.Rows.Count states =P_mat.Columns.Count ReDim A_final(1 To trades, 1 To states) ReDimstatedem(1 To states) Dim i As Long Dim totaldemand As Double Dim totaldesired As Double Dim iterations As Long iterations = 10 For i = 1 Totrades For j = 1 To states statedem(j) = A_mat(i, j) + statedem(j)A_final(i, j) = A_mat(i, j) Next j Next i For i = 1 To statestotaldemand = totaldemand + statedem(i) Next i For i = 1 To iterationsFor j = 1 To trades For z = 1 To states If A_mat(j, z) = 0 Thentotaldemand = totaldemand − A_final(j, z) statedem(z) = statedem(z) −A_final(j, z) tempalloc = A_final(j, z) A_final(j, z) =stateall(totaldemand, P_mat(j, z), statedem(z)) totaldemand = A_final(j,z) + totaldemand statedem(z) = A_final(j, z) + statedem(z) End If Next zNext j Next i allocatetrades = A_final End Function Functionstateall(totdemex, despaystate, totstateex) Dim sol1 As Double sol1 =(−(totdemex − despaystate) + ((totdemex − despaystate) {circumflex over( )} 2 + 4 * despaystate * totstateex) {circumflex over ( )} 0.5) / 2stateall = sol1 End FunctionFor this example involving two states and four traders, use of thecomputer code represented in Table 1 produces an investment amountmatrix A, as follows:

$A = \begin{matrix}5 & 10 \\7 & 0 \\1.1574 & 1.6852 \\2.8935 & 0\end{matrix}$

The matrix of unit payouts, Π, can be computed from A as described aboveand is equal to:

$\Pi = \begin{matrix}1.728 & 0 \\0 & 2.3736\end{matrix}$

The resulting payout matrix P is the product of A and Π and is equal to:

$P = \begin{matrix}8.64 & 23.7361 \\12.0961 & 0 \\2 & 4 \\5 & 0\end{matrix}$

It can be noted that the sum of each column of P, above is equal to27.7361, which is equal (in dollars) to the total amount invested so, asdesired in this example, the group of DBAR contingent claims isself-financing. The allocation is said to be in equilibrium, since theamounts invested by traders 1 and 2 are undisturbed, and traders 3 and 4receive their desired payouts, as specified above, should each stateoccur.

2.6 Interest Considerations

When investing in a group of DBAR contingent claims, traders willtypically have outstanding balances invested for periods of time and mayalso have outstanding loans or margin balances from the exchange forperiods of time. Traders will typically be paid interest on outstandinginvestment balances and typically will pay interest on outstandingmargin loans. In preferred embodiments, the effect of trade balanceinterest and margin loan interest can be made explicit in the payouts,although in alternate preferred embodiments these items can be handledoutside of the payout structure, for example, by debiting and creditinguser accounts. So, if a fraction β of a trade of one value unit is madewith cash and the rest on margin, the unit payout π_(i) in the eventthat state i occurs can be expressed as follows:

$\pi_{i} = {\frac{\left( {1 - f} \right)*T}{T_{i}} + {\beta*\left( c_{r} \right)*t_{b}} - {\left( {1 - \beta} \right)*\left( c_{p} \right)*t_{l}}}$

where the last two terms express the respective credit for tradebalances per unit invested for time t_(b) and debit for margin loans perunit invested for time t₁.

2.7 Returns and Probabilities

In a preferred embodiment of a group of DBAR contingent claims with acanonical DRF, returns which represent the percentage return per unit ofinvestment are closely related to payouts. Such returns are also closelyrelated to the notion of a financial return familiar to investors. Forexample, if an investor has purchased a stock for $100 and sells it for$110, then this investor has realized a return of 10% (and a payout of$110).

In a preferred embodiment of a group of DBAR contingent claims with acanonical DRF, the unit return, r_(i), should state i occur may beexpressed as follows:

$r_{i} = \frac{{\left( {1 - f} \right)*{\sum\limits_{i = {1\mspace{11mu} \ldots \mspace{14mu} n}}T_{i}}} - T_{i}}{T_{i}}$

if state i occurs

-   -   r_(i)=−1 otherwise, i.e., if state i does not occur

In such an embodiment, the return per unit investment in a state thatoccurs is a function of the amount invested in that state, the amountinvested in all the other states and the exchange fee. The unit returnis −100% for a state that does not occur, i.e., the entire amountinvested in the expectation of receiving a return if a state occurs isforfeited if that state fails to occur. A—100% return in such an eventhas the same return profile as, for example, a traditional optionexpiring “out of the money.” When a traditional option expires out ofthe money, the premium decays to zero, and the entire amount invested inthe option is lost.

For purposes of this specification, a payout is defined as one plus thereturn per unit invested in a given state multiplied by the amount thathas been invested in that state. The sum of all payouts P_(s) for agroup of DBAR contingent claims corresponding to all n possible statescan be expressed as follows:

${P_{s} = {{\left( {1 + r_{i}} \right)*T_{i}} + {\sum\limits_{j,{j \neq i}}{\left( {1 + r_{j}} \right)*T_{j}\mspace{14mu} 1}}}},{j = {1\mspace{14mu} \ldots \mspace{14mu} n}}$

In a preferred embodiment employing a canonical DRF, the payout P_(s)may be found for the occurrence of state i by substituting the aboveexpressions for the unit return in any state:

$\begin{matrix}{p_{S} = {{\left( {\frac{{\left( {1 - f} \right)*{\sum\limits_{i = {1\mspace{14mu} \ldots \mspace{14mu} n}}T_{i}}} - T_{i}}{T_{i}} + 1} \right)*T_{i}} + {\sum\limits_{j,{j \neq i}}{\left( {{- 1} + 1} \right)*T_{j}}}}} \\{= {\left( {1 - f} \right)*{\sum\limits_{i = {1\mspace{14mu} \ldots \mspace{14mu} n}}T_{i}}}}\end{matrix}$

Accordingly, in such a preferred embodiment, for the occurrence of anygiven state, no matter what state, the aggregate payout to all of thetraders as a whole is one minus the transaction fee paid to the exchange(expressed in this preferred embodiment as a percentage of totalinvestment across all the states), multiplied by the total amountinvested across all the states for the group of DBAR contingent claims.This means that in a preferred embodiment of a group of the DBARcontingent claims, and assuming no credit or similar risks to theexchange, (1) the exchange has zero probability of loss in any givenstate; (2) for the occurrence of any given state, the exchange receivesan exchange fee and is not exposed to any risk; (3) payouts and returnsare a function of demand flow, i.e., amounts invested; and (4)transaction fees or exchange fees can be a simple function of aggregateamount invested.

Other transaction fees can be implemented. For example, the transactionfee might have a fixed component for some level of aggregate amountinvested and then have either a sliding or fixed percentage applied tothe amount of the investment in excess of this level. Other methods fordetermining the transaction fee are apparent to those of skill in theart, from this specification or based on practice of the presentinvention.

In a preferred embodiment, the total distribution of amounts invested inthe various states also implies an assessment by all traderscollectively of the probabilities of occurrence of each state. In apreferred embodiment of a group of DBAR contingent claims with acanonical DRF, the expected return E(r_(i)) for an investment in a givenstate i (as opposed to the return actually received once outcomes areknown) may be expressed as the probability weighted sum of the returns:

E(r _(i))=q _(i) *r _(i)+(1−q _(i))*−1=q _(i)*(1+r _(i))−1

Where q_(i) is the probability of the occurrence of state i implied bythe matrix A (which contains all of the invested amounts for all statesin the group of DBAR contingent claims). Substituting the expression forr_(i) from above yields:

${E\left( r_{i} \right)} = {{q_{i}*\left( \frac{\left( {1 - f} \right)*{\sum\limits_{i}T_{i}}}{T_{i}} \right)} - 1}$

In an efficient market, the expected return E(r_(i)) across all statesis equal to the transaction costs of trading, i.e., on average, alltraders collectively earn returns that do not exceed the costs oftrading. Thus, in an efficient market for a group of DBAR contingentclaims using a canonical, where E(r_(i)) equals the transaction fee, -f,the probability of the occurrence of state i implied by matrix A iscomputed to be:

$q_{i} = \frac{T_{i}}{\sum\limits_{i}T_{i}}$

Thus, in such a group of DBAR contingent claims, the implied probabilityof a given state is the ratio of the amount invested in that statedivided by the total amount invested in all states. This relationshipallows traders in the group of DBAR contingent claims (with a canonicalDRF) readily to calculate the implied probability which traders attachto the various states.

Information of interest to a trader typically includes the amountsinvested per state, the unit return per state, and implied stateprobabilities. An advantage of the DBAR exchange of the presentinvention is the relationship among these quantities. In a preferredembodiment, if the trader knows one, the other two can be readilydetermined. For example, the relationship of unit returns to theoccurrence of a state and the probability of the occurrence of thatstate implied by A can be expressed as follows:

$q_{i} = \frac{\left( {1 - f} \right)}{\left( {1 + r_{i}} \right)}$

The expressions derived above show that returns and implied stateprobabilities may be calculated from the distribution of the investedamounts, T_(i), for all states, i=1 . . . n. In the traditional markets,the amount traded across the distribution of states (limit order book),is not readily available. Furthermore, in traditional markets there areno such ready mathematical calculations that relate with any precisioninvested amounts or the limit order book to returns or prices whichclear the market, i.e., prices at which the supply equals the demand.Rather, in the traditional markets, specialist brokers and market makerstypically have privileged access to the distribution of bids and offers,or the limit order book, and frequently use this privileged informationin order to set market prices that balance supply and demand at anygiven time in the judgment of the market maker.

2.8 Computations When Invested Amounts Are Large

In a preferred embodiment of a group of DBAR contingent claims using acanonical DRF, when large amounts are invested across the distributionof states, it may be possible to perform approximate investmentallocation calculations in order to generate desired payoutdistributions. The payout, p, should state i occur for a trader whoconsiders making an investment of size a in state i has been shown aboveto be:

$p = {\left( \frac{T + \alpha}{T_{i} + \alpha} \right)*\alpha}$

If α is small compared to both the total invested in state i and thetotal amount invested in all the states, then adding a to state i willnot have a material effect on the ratio of the total amount invested inall the states to the total amount invested in state i. In thesecircumstances,

$\frac{T + \alpha}{T_{i} + \alpha} \approx \frac{T}{T_{i}}$

Thus, in preferred embodiments where an approximation is acceptable, thepayout to state i may be expressed as:

$p \approx {\frac{T}{T_{i}}*\alpha}$

In these circumstances, the investment needed to generate the payout pis:

${\alpha \approx {\frac{T_{i}}{T}*p}} = {q_{i}*p}$

These expressions indicate that in preferred embodiments, the amount tobe invested to generate a desired payout is approximately equal to theratio of the total amount invested in state i to the total amountinvested in all states, multiplied by the desired payout. This isequivalent to the implied probability multiplied by the desired payout.Applying this approximation to the expression CDRF 2, above, yields thefollowing:

A≈P*Π ⁻¹ =P*Q

where the matrix Q, of dimension n×n, is equal to the inverse of unitpayouts n, and has along the diagonals q_(i) for i=1 . . . n. Thisexpression provides an approximate but more readily calculable solutionto CDRF 2 as the expression implicitly assumes that an amount investedby a trader has approximately no effect on the existing unit payouts orimplied probabilities. This approximate solution, which is linear andnot quadratic, will sometimes be used in the following examples where itcan be assumed that the total amounts invested are large in relation toany given trader's particular investment.

3. Examples of Groups of DBAR Contingent Claims

3.1 DBAR Range Derivatives

A DBAR Range Derivative (DBAR RD) is a type of group of DBAR contingentclaims implemented using a canonical DRF described above (although aDBAR range derivative can also be implemented, for example, for a groupof DBAR contingent claims, including DBAR digital options, based on thesame ranges and economic events established below using, e.g., anon-canonical DRF and an OPF). In a DBAR RD, a range of possibleoutcomes associated with an observable event of economic significance ispartitioned into defined states. In a preferred embodiment, the statesare defined as discrete ranges of possible outcomes so that the entiredistribution of states covers all the possible outcomes—that is, thestates are collectively exhaustive. Furthermore, in a DBAR RD, statesare preferably defined so as to be mutually exclusive as well, meaningthat the states are defined in such a way so that exactly one stateoccurs. If the states are defined to be both mutually exclusive andcollectively exhaustive, the states form the basis of a probabilitydistribution defined over discrete outcome ranges. Defining the statesin this way has many advantages as described below, including theadvantage that the amount which traders invest across the states can bereadily converted into implied probabilities representing the collectiveassessment of traders as to the likelihood of the occurrence of eachstate.

The system and methods of the present invention may also be applied todetermine projected DBAR RD returns for various states at the beginningof a trading period. Such a determination can be, but need not be, madeby an exchange. In preferred embodiments of a group of DBAR contingentclaims the distribution of invested amounts at the end of a tradingperiod determines the returns for each state, and the amount invested ineach state is a function of trader preferences and probabilityassessments of each state. Accordingly, some assumptions typically needto be made in order to determine preliminary or projected returns foreach state at the beginning of a trading period.

An illustration is provided to explain further the operation of DBARRDs. In the following illustration, it is assumed that all traders arerisk neutral so that implied probabilities for a state are equal to theactual probabilities, and so that all traders have identical probabilityassessments of the possible outcomes for the event defining thecontingent claim. For convenience in this illustration, the eventforming the basis for the contingent claims is taken to be a closingprice of a security, such as a common stock, at some future date; andthe states, which represent the possible outcomes of the level of theclosing price, are defined to be distinct, mutually exclusive andcollectively exhaustive of the range of (possible) closing prices forthe security. In this illustration, the following notation is used:

-   -   τ represents a given time during the trading period at which        traders are making investment decisions    -   θ represents the time corresponding to the expiration of the        contingent claim    -   V_(τ) represents the price of underlying security at time τ    -   V_(θ) represents the price of underlying security at time θ    -   Z(τ,θ) represents the present value of one unit of value payable        at time θ evaluated at time    -   D(τ,θ) represents dividends or coupons payable between time τ        and θ    -   σ_(t) represents annualized volatility of natural logarithm        returns of the underlying security    -   dz represents the standard normal variate        Traders make choices at a representative time, τ, during a        trading period which is open, so that time τ is temporally        subsequent to the current trading period's TSD.

In this illustration, and in preferred embodiments, the defined statesfor the group of contingent claims for the final closing price V_(θ) areconstructed by discretizing the full range of possible prices intopossible mutually exclusive and collectively exhaustive states. Thetechnique is similar to forming a histogram for discrete countable data.The endpoints of each state can be chosen, for example, to be equallyspaced, or of varying spacing to reflect the reduced likehood of extremeoutcomes compared to outcomes near the mean or median of thedistribution. States may also be defined in other manners apparent toone of skill in the art. The lower endpoint of a state can be includedand the upper endpoint excluded, or vice versa. In any event, inpreferred embodiments, the states are defined (as explained below) tomaximize the attractiveness of investment in the group of DBARcontingent claims, since it is the invested amounts that ultimatelydetermine the returns that are associated with each defined state.

The procedure of defining states, for example for a stock price, can beaccomplished by assuming lognormality, by using statistical estimationtechniques based on historical time series data and cross-section marketdata from options prices, by using other statistical distributions, oraccording to other procedures known to one of skill in the art orlearned from this specification or through practice of the presentinvention. For example, it is quite common among derivatives traders toestimate volatility parameters for the purpose of pricing options byusing the econometric techniques such as GARCH. Using these parametersand the known dividend or coupons over the time period from τ to θ, forexample, the states for a DBAR RD can be defined.

A lognormal distribution is chosen for this illustration since it iscommonly employed by derivatives traders as a distributional assumptionfor the purpose of evaluating the prices of options and other derivativesecurities. Accordingly, for purposes of this illustration it is assumedthat all traders agree that the underlying distribution of states forthe security are lognormally distributed such that:

${\overset{\sim}{V}}_{\theta} = {\left( {\frac{V_{\tau}}{Z\left( {\tau,\theta} \right)} - \frac{D\left( {\tau,\theta} \right)}{Z\left( {\tau,\theta} \right)}} \right)*^{{{- \sigma^{2}}/2}*{({\theta - \tau})}}*^{\sigma*\sqrt{\theta - \tau}*{dz}}}$

where the “tilde” on the left-hand side of the expression indicates thatthe final closing price of the value of the security at time θ is yet tobe known. Inversion of the expression for dz and discretization ofranges yields the following expressions:

${dz} = {{\ln\left( \frac{V_{\theta}*^{\frac{\sigma^{2}}{2}*{({\theta - \tau})}}}{\left( {\frac{V_{\tau}}{Z\left( {\tau,\theta} \right)} - \frac{D\left( {\tau,\theta} \right)}{Z\left( {\tau,\theta} \right)}} \right)} \right)}/\left( {\sigma*\sqrt{\theta - \tau}} \right)}$q_(i)(V_(i) <  = V_(θ) < V_(i + 1)) = cdf(dz_(i + 1)) − cdf(dz_(i))${r_{i}\left( {V_{i}<=V_{\theta} < V_{i + 1}} \right)} = {\frac{\left( {1 - f} \right)}{q_{i}\left( {V_{i}<=V_{\theta} < V_{i + 1}} \right)} - 1}$

where cdf(dz) is the cumulative standard normal function.

The assumptions and calculations reflected in the expressions presentedabove can also be used to calculate indicative returns (“openingreturns”), r_(i) at a beginning of the trading period for a given groupof DBAR contingent claims. In a preferred embodiment, the calculatedopening returns are based on the exchange's best estimate of theprobabilities for the states defining the claim and therefore mayprovide good indications to traders of likely returns once trading isunderway. In another preferred embodiment, described with respect toDBAR digital options in Section 6 and another embodiment described inSection 7, a very small number of value units may be used in each stateto initialize the contract or group of contingent claims. Of course,opening returns need not be provided at all, as traded amounts placedthroughout the trading period allows the calculation of actual expectedreturns at any time during the trading period.

The following examples of DBAR range derivatives and other contingentclaims serve to illustrate their operation, their usefulness inconnection with a variety of events of economic significance involvinginherent risk or uncertainty, the advantages of exchanges for groups ofDBAR contingent claims, and, more generally, systems and methods of thepresent invention. Sections 6 and 7 also provide examples of DBARcontingent claims of the present invention that provide profit and lossscenarios comparable to those provided by digital options inconventional options markets, and that can be based on any of thevariety of events of economic signficance described in the followingexamples of DBAR RDs.

In each of the examples in this Section, a state is defined to include arange of possible outcomes of an event of economic significance. Theevent of economic significance for any DBAR auction or market (includingany market or auction for DBAR digital options) can be, for example, anunderlying economic event (e.g., price of stock) or a measured parameterrelated to the underlying economic event (e.g., a measured volatility ofthe price of stock). A curved brace “(“or”)” denotes strict inequality(e.g., “greater than” or “less than,” respectively) and a square brace“]” or “[” shall denote weak inequality (e.g., “less than or equal to”or “greater than or equal to,” respectively). For simplicity, and unlessotherwise stated, the following examples also assume that the exchangetransaction fee, f, is zero.

Example 3.1.1 DBAR Contingent Claim On Underlying Common Stock

-   -   Underlying Security: Microsoft Corporation Common Stock (“MSFT”)    -   Date: Aug. 18, 1999    -   Spot Price: 85    -   Market Volatility: 50% annualized    -   Trading Start Date: Aug. 18, 1999, Market Open    -   Trading End Date: Aug. 18, 1999, Market Close    -   Expiration: Aug. 19, 1999, Market Close    -   Event: MSFT Closing Price at Expiration    -   Trading Time: 1 day    -   Duration to TED: 1 day    -   Dividends Payable to Expiration: 0    -   Interbank short-term interest rate to Expiration: 5.5%        (Actual/360 daycount)    -   Present Value factor to Expiration: 0.999847    -   Investment and Payout Units: U.S. Dollars (“USD”)

In this Example 3.1.1, the predetermined termination criteria are theinvestment in a contingent claim during the trading period and theclosing of the market for Microsoft common stock on Aug. 19, 1999.

If all traders agree that the underlying distribution of closing pricesis lognormally distributed with volatility of 50%, then an illustrative“snapshot” distribution of invested amounts and returns for $100 millionof aggregate investment can be readily calculated to yield the followingtable.

TABLE 3.1.1-1 States Investment in State (′000) Return Per Unit if StateOccurs (0, 80] 1,046.58 94.55 (80, 80.5] 870.67 113.85 (80.5, 81]1,411.35 69.85 (81, 81.5] 2,157.85 45.34 (81.5, 82] 3,115.03 31.1 (82,82.5] 4,250.18 22.53 (82.5, 83] 5,486.44 17.23 (83, 83.5] 6,707.18 13.91(83.5, 84] 7,772.68 11.87 (84, 84.5] 8,546.50 10.7 (84.5, 85] 8,924.7110.2 (85, 85.5] 8,858.85 10.29 (85.5, 86] 8,366.06 10.95 (86, 86.5]7,523.13 12.29 (86.5, 87] 6,447.26 14.51 (87, 87.5] 5,270.01 17.98(87.5, 88] 4,112.05 23.31 (88, 88.5] 3,065.21 31.62 (88.5, 89] 2,184.544.78 (89, 89.5] 1,489.58 66.13 (89.5, 90] 972.56 101.82 (90, ∞]1,421.61 69.34

Consistent with the design of a preferred embodiment of a group of DBARcontingent claims, the amount invested for any given state is inverselyrelated to the unit return for that state.

In preferred embodiments of groups of DBAR contingent claims, traderscan invest in none, one or many states. It may be possible in preferredembodiments to allow traders efficiently to invest in a set, subset orcombination of states for the purposes of generating desireddistributions of payouts across the states. In particular, traders maybe interested in replicating payout distributions which are common inthe traditional markets, such as payouts corresponding to a long stockposition, a short futures position, a long option straddle position, adigital put or digital call option.

If in this Example 3.1.1 a trader desired to hedge his exposure toextreme outcomes in MSFT stock, then the trader could invest in statesat each end of the distribution of possible outcomes. For instance, atrader might decide to invest $100,000 in states encompassing pricesfrom $0 up to and including $83 (i.e., (0,83]) and another $100,000 instates encompassing prices greater than $86.50 (i.e., (86.5,∞]). Thetrader may further desire that no matter what state actually occurswithin these ranges (should the state occur in either range) upon thefulfillment of the predetermined termination criteria, an identicalpayout will result. In this Example 3.1.1, a multi-state investment iseffectively a group of single state investments over each multi-staterange, where an amount is invested in each state in the range inproportion to the amount previously invested in that state. If, forexample, the returns provided in Table 3.1.1-1 represent finalizedprojected returns at the end of the trading period, then eachmulti-state investment may be allocated to its constituent states on apro-rata or proportional basis according to the relative amountsinvested in the constituent states at the close of trading. In this way,more of the multi-state investment is allocated to states with largerinvestments and less allocated to the states with smaller investments.

Other desired payout distributions across the states can be generated byallocating the amount invested among the constituent states in differentways so as achieve a trader's desired payout distribution. A trader mayselect, for example, both the magnitude of the payouts and how thosepayouts are to be distributed should each state occur and let the DBARexchange's multi-state allocation methods determine (1) the size of theamount invested in each particular constituent state; (2) the states inwhich investments will be made, and (3) how much of the total amount tobe invested will be invested in each of the states so determined. Otherexamples below demonstrate how such selections may be implemented.

Since in preferred embodiments the final projected returns are not knownuntil the end of a given trading period, in such embodiments a previousmulti-state investment is reallocated to its constituent statesperiodically as the amounts invested in each state (and thereforereturns) change during the trading period. At the end of the tradingperiod when trading ceases and projected returns are finalized, in apreferred embodiment a final reallocation is made of all the multi-stateinvestments. In preferred embodiments, a suspense account is used torecord and reallocate multi-state investments during the course oftrading and at the end of the trading period.

Referring back to the illustration assuming two multi-state trades overthe ranges (0,83] and (86.5,∞] for MSFT stock, Table 3.1.1-2 shows howthe multi-state investments in the amount of $100,000 each could beallocated according to a preferred embodiment to the individual statesover each range in order to achieve a payout for each multi-state rangewhich is identical regardless of which state occurs within each range.In particular, in this illustration the multi-state investments areallocated in proportion to the previously invested amount in each state,and the multi-state investments marginally lower returns over (0,83] and(86.5,∞], but marginally increase returns over the range (83, 86.5], asexpected.

To show that the allocation in this example has achieved its goal ofdelivering the desired payouts to the trader, two payouts for the (0,83] range are considered. The payout, if constituent state (80.5, 81]occurs, is the amount invested in that state ($7.696) multiplied by oneplus the return per unit if that state occurs, or(1+69.61)*7.696=$543.40. A similar analysis for the state (82.5, 83]shows that, if it occurs, the payout is equal to(1+17.162)*29.918=$543.40. Thus, in this illustration, the traderreceives the same payout no matter which constituent state occurs withinthe multi-state investment. Similar calculations can be performed forthe range [86.5,∞]. For example, under the same assumptions, the payoutfor the constituent state [86.5,87] would receive a payout of $399.80 ifthe stock price fill in that range after the fulfillment of all of thepredetermined termination criteria. In this illustration, eachconstituent state over the range [86.5, ∞] would receive a payout of$399.80, no matter which of those states occurs.

TABLE 3.1.1-2 Traded Amount in State Return Per Unit Multi-State States(′000) if State Occurs Allocation (′000) (0, 80] 1052.29 94.22 5.707(80, 80.5] 875.42 113.46 4.748 (80.5, 81] 1,419.05 69.61 7.696 (81,81.5] 2,169.61 45.18 11.767 (81.5, 82] 3,132.02 30.99 16.987 (82, 82.5]4,273.35 22.45 23.177 (82.5, 83] 5,516.36 17.16 29.918 (83, 83.5]6,707.18 13.94 (83.5, 84] 7,772.68 11.89 (84, 84.5] 8,546.50 10.72(84.5, 85] 8,924.71 10.23 (85, 85.5] 8,858.85 10.31 (85.5, 86] 8,366.0610.98 (86, 86.5] 7,523.13 12.32 (86.5, 87] 6,473.09 14.48 25.828 (87,87.5] 5,291.12 17.94 21.111 (87.5, 88] 4,128.52 23.27 16.473 (88, 88.5]3,077.49 31.56 12.279 (88.5, 89] 2,193.25 44.69 8.751 (89, 89.5]1,495.55 66.00 5.967 (89.5, 90] 976.46 101.62 3.896 (90, ∞] 1,427.3169.20 5.695

Options on equities and equity indices have been one of the moresuccessful innovations in the capital markets. Currently, listed optionsproducts exist for various underlying equity securities and indices andfor various individual option series. Unfortunately, certain marketslack liquidity. Specifically, liquidity is usually limited to only ahandful of the most widely recognized names. Most option markets areessentially dealer-based. Even for options listed on an exchange,market-makers who stand ready to buy or sell options across all strikesand maturities are a necessity. Although market participants trading aparticular option share an interest in only one underlying equity, theexistence of numerous strike prices scatters liquidity coming into themarket thereby making dealer support essential. In all but the mostliquid and active exchange-traded options, chances are rare that twooption orders will meet for the same strike, at the same price, at thesame time, and for the same volume. Moreover, market-makers in listedand over-the-counter (OTC) equities must allocate capital and managerisk for all their positions. Consequently, the absolute amount ofcapital that any one market-maker has on hand is naturally constrainedand may be insufficient to meet the volume of institutional demand.

The utility of equity and equity-index options is further constrained bya lack of transparency in the OTC markets. Investment banks typicallyoffer customized option structures to satisfy their customers.Customers, however, are sometimes hesitant to trade in environmentswhere they have no means of viewing the market and so are uncertainabout getting the best prevailing price.

Groups of DBAR contingent claims can be structured using the system andmethods of the present invention to provide market participants with afuller, more precise view of the price for risks associated with aparticular equity.

Example 3.1.2 Multiple Multi-State Investments

If numerous multi-state investments are made for a group of DBARcontingent claims, then in a preferred embodiment an iterative procedurecan be employed to allocate all of the multi-state investments to theirrespective constituent states. In preferred embodiments, the goal wouldbe to allocate each multi-state investment in response to changes inamounts invested during the trading period, and to make a finalallocation at the end of the trading period so that each multi-stateinvestment generates the payouts desired by the respective trader. Inpreferred embodiments, the process of allocating multi-state investmentscan be iterative, since allocations depend upon the amounts tradedacross the distribution of states at any point in time. As aconsequence, in preferred embodiments, a given distribution of investedamounts will result in a certain allocation of a multi-state investment.When another multi-state investment is allocated, the distribution ofinvested amounts across the defined states may change and thereforenecessitate the reallocation of any previously allocated multi-stateinvestments. In such preferred embodiments, each multi-state allocationis re-performed so that, after a number of iterations through all of thepending multi-state investments, both the amounts invested and theirallocations among constituent states in the multi-state investments nolonger change with each successive iteration and a convergence isachieved. In preferred embodiments, when convergence is achieved,further iteration and reallocation among the multi-state investments donot change any multi-state allocation, and the entire distribution ofamounts invested across the states remains stable and is said to be inequilibrium. Computer code, as illustrated in Table 1 above or relatedcode readily apparent to one of skill in the art, can be used toimplement this iterative procedure.

A simple example demonstrates a preferred embodiment of an iterativeprocedure that may be employed. For purposes of this example, apreferred embodiment of the following assumptions are made: (i) thereare four defined states for the group of DBAR contingent claims; (ii)prior to the allocation of any multi-state investments, $100 has beeninvested in each state so that the unit return for each of the fourstates is 3; (iii) each desires that each constituent state in amulti-state investment provides the same payout regardless of whichconstituent state actually occurs; and (iv) that the following othermulti-state investments have been made:

TABLE 3.1.2-1 Investment Invested Number State 1 State 2 State 3 State 4Amount, $ 1001 X X 0 0 100 1002 X 0 X X 50 1003 X X 0 0 120 1004 X X X 0160 1005 X X X 0 180 1006 0 0 X X 210 1007 X X X 0 80 1008 X 0 X X 9501009 X X X 0 1000 1010 X X 0 X 500 1011 X 0 0 X 250 1012 X X 0 0 1001013 X 0 X 0 500 1014 0 X 0 X 1000 1015 0 X X 0 170 1016 0 X 0 X 1201017 X 0 X 0 1000 1018 0 0 X X 200 1019 X X X 0 250 1020 X X 0 X 3001021 0 X X X 100 1022 X 0 X X 400where an “X” in each state represents a constituent state of themulti-state trade. Thus, as depicted in Table 3.1.2-1, trade number 1001in the first row is a multi-state investment of $100 to be allocatedamong constituent states 1 and 2, trade number 1002 in the second row isanother multi-state investment in the amount of $50 to be allocatedamong constituent states 1, 3, and 4; etc.

Applied to the illustrative multi-state investment described above, theiterative procedure described above and embodied in the illustrativecomputer code in Table 1, results in the following allocations:

TABLE 3.1.2-2 Investment Number State 1($) State 2($) State 3($) State4($) 1001 73.8396 26.1604 0 0 1002 26.66782 0 12.53362 10.79856 100388.60752 31.39248 0 0 1004 87.70597 31.07308 41.22096 0 1005 98.6692134.95721 46.37358 0 1006 0 0 112.8081 97.19185 1007 43.85298 15.5365420.61048 0 1008 506.6886 0 238.1387 205.1726 1009 548.1623 194.2067257.631 0 1010 284.2176 100.6946 0 115.0878 1011 177.945 0 0 72.055 101273.8396 26.1604 0 0 1013 340.1383 0 159.8617 0 1014 0 466.6488 0533.3512 1015 0 73.06859 96.93141 0 1016 0 55.99785 0 64.00215 1017680.2766 0 319.7234 0 1018 0 0 107.4363 92.56367 1019 137.0406 48.5516864.40774 0 1020 170.5306 60.41675 0 69.05268 1021 0 28.82243 38.2352932.94229 1022 213.3426 0 100.2689 86.38848In Table 3.1.2-2 each row shows the allocation among the constituentstates of the multi-state investment entered into the corresponding rowof Table 3.1.2-1, the first row of Table 3.1.2-2 that investment number1001 in the amount of $100 has been allocated $73.8396 to state 1 andthe remainder to state 2.

It may be shown that the multi-state allocations identified above resultin payouts to traders which are desired by the traders—that is, in thisexample the desired payouts are the same regardless of which stateoccurs among the constituent states of a given multi-state investment.Based on the total amount invested as reflected in Table 3.1.2-2 andassuming a zero transaction fee, the unit returns for each state are:

State 1 State 2 State 3 State 4 Return Per Dollar 1.2292 5.2921 3.74314.5052 InvestedConsideration of Investment 1022 in this example, illustrates theuniformity of payouts for each state in which an investment is made(i.e., states 1, 3 and 4). If state 1 occurs, the total payout to thetrader is the unit return for state 1—1.2292—multiplied by the amounttraded for state 1 in trade 1022—$213.3426—plus the initialtrade—$213.3426. This equals 1.2292*213.3426+213.3426=$475.58. If state3 occurs, the payout is equal to 3.7431*100.2689+100.2689=$475.58.Finally, if state 4 occurs, the payout is equal to4.5052*86.38848+86.38848=$475.58. So a preferred embodiment of amulti-state allocation in this example has effected an allocation amongthe constituent states so that (1) the desired payout distributions inthis example are achieved, i.e., payouts to constituent states are thesame no matter which constituent state occurs, and (2) furtherreallocation iterations of multi-state investments do not change therelative amounts invested across the distribution of states for all themulti-state trades.

Example 3.1.3 Alternate Price Distributions

Assumptions regarding the likely distribution of traded amounts for agroup of DBAR contingent claims may be used, for example, to computereturns for each defined state per unit of amount invested at thebeginning of a trading period (“opening returns”). For various reasons,the amount actually invested in each defined state may not reflect theassumptions used to calculate the opening returns. For instance,investors may speculate that the empirical distribution of returns overthe time horizon may differ from the no-arbitrage assumptions typicallyused in option pricing. Instead of a lognormal distribution, moreinvestors might make investments expecting returns to be significantlypositive rather than negative (perhaps expecting favorable news). InExample 3.1.1, for instance, if traders invested more in states above$85 for the price of MSFT common stock, the returns to states below $85could therefore be significantly higher than returns to states above$85.

In addition, it is well known to derivatives traders that traded optionprices indicate that price distributions differ markedly fromtheoretical lognormality or similar theoretical distributions. Theso-called volatility skew or “smile” refers to out-of-the-money put andcall options trading at higher implied volatilities than options closerto the money. This indicates that traders often expect the distributionof prices to have greater frequency or mass at the extreme observationsthan predicted according to lognormal distributions. Frequently, thiseffect is not symmetric so that, for example, the probability of largelower price outcomes are higher than for extreme upward outcomes.

Consequently, in a group of DBAR contingent claims of the presentinvention, investment in states in these regions may be more prevalentand, therefore, finalized returns on outcomes in those regions lower.For example, using the basic DBAR contingent claim information fromExample 3.1.1, the following returns may prevail due to investorexpectations of return distributions that have more frequent occurrencesthan those predicted by a lognormal distribution, and thus are skewed tothe lower possible returns. In statistical parlance, such a distributionexhibits higher kurtosis and negative skewness in returns than theillustrative distribution used in Example 3.1.1 and reflected in Table3.1.1-1.

TABLE 3.1.3-1 DBAR Contingent Claim Returns Illustrating NegativelySkewed and Leptokurtotic Return Distribution Amount Invested in StateStates (′000) Return Per Unit if State Occurs (0, 80] 3,150 30.746 (80,80.5] 1,500 65.667 (80.5, 81] 1,600 61.5 (81, 81.5] 1,750 56.143 (81.5,82] 2,100 46.619 (82, 82.5] 2,550 38.216 (82.5, 83] 3,150 30.746 (83,83.5] 3,250 29.769 (83.5, 84] 3,050 31.787 (84, 84.5] 8,800 10.363(84.5, 85] 14,300 5.993 (85, 85.5] 10,950 8.132 (85.5, 86] 11,300 7.85(86, 86.5] 10,150 8.852 (86.5, 87] 11,400 7.772 (87, 87.5] 4,550 20.978(87.5, 88] 1,350 73.074 (88, 88.5] 1,250 79.0 (88.5, 89] 1,150 85.957(89, 89.5] 700 141.857 (89.5, 90] 650 152.846 (90, ∞] 1,350 73.074

The type of complex distribution illustrated in Table 3.1.3-1 isprevalent in the traditional markets. Derivatives traders, actuaries,risk managers and other traditional market participants typically usesophisticated mathematical and analytical tools in order to estimate thestatistical nature of future distributions of risky market outcomes.These tools often rely on data sets (e.g., historical time series,options data) that may be incomplete or unreliable. An advantage of thesystems and methods of the present invention is that such analyses fromhistorical data need not be complicated, and the full outcomedistribution for a group of DBAR contingent claims based on any givenevent is readily available to all traders and other interested partiesnearly instantaneously after each investment.

Example 3.1.4 States Defined For Return Uniformity

It is also possible in preferred embodiments of the present invention todefine states for a group of DBAR contingent claims with irregular orunevenly distributed intervals, for example, to make the traded amountacross the states more liquid or uniform. States can be constructed froma likely estimate of the final distribution of invested amounts in orderto make the likely invested amounts, and hence the returns for eachstate, as uniform as possible across the distribution of states. Thefollowing table illustrates the freedom, using the event and tradingperiod from Example 3.1.1, to define states so as to promoteequalization of the amount likely to be invested in each state.

TABLE 3.1.4-1 State Definition to Make Likely Demand Uniform AcrossStates Invested Amount in States State (′000) Return Per Unit if StateOccurs (0, 81.403] 5,000 19 (81.403, 82.181] 5,000 19 (82.181, 82.71]5,000 19 (82.71, 83.132] 5,000 19 (83.132, 83.497] 5,000 19 (83.497,83.826] 5,000 19 (83.826, 84.131] 5,000 19 (84.131, 84.422] 5,000 19(84.422, 84.705] 5,000 19 (84.705, 84.984] 5,000 19 (84.984, 85.264]5,000 19 (85.264, 85.549] 5,000 19 (85.549, 85.845] 5,000 19 (85.845,86.158] 5,000 19 (86.158, 86.497] 5,000 19 (86.497, 86.877] 5,000 19(86.877, 87.321] 5,000 19 (87.321, 87.883] 5,000 19 (87.883, 88.722]5,000 19 (88.722, ∞] 5,000 19

If investor expectations coincide with the often-used assumption of thelognormal distribution, as reflected in this example, then investmentactivity in the group of contingent claims reflected in Table 3.1.4-1will converge to investment of the same amount in each of the 20 statesidentified in the table. Of course, actual trading will likely yieldfinal market returns which deviate from those initially chosen forconvenience using a lognormal distribution.

Example 3.1.5 Government Bond—Uniformly Constructed States

The event, defined states, predetermined termination criteria and otherrelevant data for an illustrative group of DBAR contingent claims basedon a U.S. Treasury Note are set forth below:

-   -   Underlying Security: United States Treasury Note, 5.5%, May 31,        2003    -   Bond Settlement Date: Jun. 25, 1999    -   Bond Maturity Date: May 31, 2003    -   Contingent Claim Expiration: 7/2/99, Market Close, 4:00 p.m. EST    -   Trading Period Start Date: Jun. 25, 1999, 4:00 p.m., EST    -   Trading Period End Date: Jun. 28, 1999, 4:00 p.m., EST    -   Next Trading Period Open: Jun. 28, 1999, 4:00 p.m., EST    -   Next Trading Period Close Jun. 29, 1999, 4:00 p.m., EST    -   Event: Closing Composite Price as reported on Bloomberg at Claim        Expiration    -   Trading Time: 1 day    -   Duration from TED: 5 days    -   Coupon: 5.5%    -   Payment Frequency: Semiannual    -   Daycount Basis Actual/Actual    -   Dividends Payable over Time Horizon: 2.75 per 100 on Jun. 30,        1999    -   Treasury note repo rate over Time Horizon: 4.0% (Actual/360        daycount)    -   Spot Price: 99.8125    -   Forward Price at Expiration: 99.7857    -   Price Volatility: 4.7%    -   Trade and Payout Units: U.S. Dollars    -   Total Demand in Current Trading Period: $50 million    -   Transaction Fee: 25 basis points (0.0025%)

TABLE 3.1.5-1 DBAR Contingent Claims on U.S. Government Note StatesInvestment in State ($) Unit Return if State Occurs (0, 98] 139690.1635356.04 (98, 98.25] 293571.7323 168.89 (98.25, 98.5] 733769.9011 66.97(98.5, 98.75] 1574439.456 30.68 (98.75, 99] 2903405.925 16.18 (99, 99.1]1627613.865 29.64 (99.1, 99.2] 1914626.631 25.05 (99.2, 99.3]2198593.057 21.68 (99.3, 99.4] 2464704.885 19.24 (99.4, 99.5]2697585.072 17.49 (99.5, 99.6] 2882744.385 16.30 (99.6, 99.7]3008078.286 15.58 (99.7, 99.8] 3065194.576 15.27 (99.8, 99.9]3050276.034 15.35 (99.9, 100] 2964602.039 15.82 (100, 100.1] 2814300.65716.72 (100.1, 100.2] 2609637.195 18.11 (100.2, 100.3] 2363883.036 20.10(100.3, 100.4] 2091890.519 22.84 (100.4, 100.5] 1808629.526 26.58(100.5, 100.75] 3326547.254 13.99 (100.75, 101] 1899755.409 25.25 (101,101.25] 941506.1374 51.97 (101.25, 101.5] 405331.6207 122.05 (101.5, ∞]219622.6373 226.09

This Example 3.1.5 and Table 3.1.5-1 illustrate how readily the methodsand systems of the present invention may be adapted to sources of risk,whether from stocks, bonds, or insurance claims. Table 3.1.5-1 alsoillustrates a distribution of defined states which is irregularlyspaced—in this case finer toward the center of the distribution andcoarser at the ends—in order to increase the amount invested in theextreme states.

Example 3.1.6 Outperformance Asset Allocation—Uniform Range

One of the advantages of the system and methods of the present inventionis the ability to construct groups of DBAR contingent claims based onmultiple events and their inter-relationships. For example, many indexfund money managers often have a fundamental view as to whether indicesof high quality fixed income securities will outperform major equityindices. Such opinions normally are contained within a manager's modelfor allocating funds under management between the major asset classessuch as fixed income securities, equities, and cash.

This Example 3.1.6 illustrates the use of a preferred embodiment of thesystems and methods of the present invention to hedge the real-worldevent that one asset class will outperform another. The illustrativedistribution of investments and calculated opening returns for the groupof contingent claims used in this example are based on the assumptionthat the levels of the relevant asset-class indices are jointlylognormally distributed with an assumed correlation. By defining a groupof DBAR contingent claims on a joint outcome of two underlying events,traders are able to express their views on the co-movements of theunderlying events as captured by the statistical correlation between theevents. In this example, the assumption of a joint lognormaldistribution means that the two underlying events are distributed asfollows:

$\mspace{79mu} {{\overset{\sim}{V}}_{\theta}^{1} = {\left( {\frac{V_{\tau}^{1}}{Z^{1}\left( {\tau,\theta} \right)} - \frac{D^{1}\left( {\tau,\theta} \right)}{Z^{1}\left( {\tau,\theta} \right)}} \right)*^{{{- \sigma_{1}^{2}}/2}*{({\theta - \tau})}}*^{\sigma_{1}*\sqrt{\theta - \tau}*{dz}_{1}}}}$$\mspace{79mu} {{\overset{\sim}{V}}_{\theta}^{2} = {\left( {\frac{V_{\tau}^{2}}{Z^{2}\left( {\tau,\theta} \right)} - \frac{D^{2}\left( {\tau,\theta} \right)}{Z^{2}\left( {\tau,\theta} \right)}} \right)*^{{{- \sigma_{2}^{2}}/2}*{({\theta - \tau})}}*^{\sigma_{2}*\sqrt{\theta - \tau}*{dz}_{2}}}}$${g\left( {{dz}_{1},{dz}_{2}} \right)} = {\frac{1}{2*\pi*\sqrt{1 - \rho^{2}}}*{\exp\left( {- \frac{\left( {{dz}_{1}^{2} + {dz}_{2}^{2} - {2*\rho*{dz}_{1}*{dz}_{1}}} \right)}{2*\left( {1 - \rho^{2}} \right)}} \right)}}$

where the subscripts and superscripts indicate each of the two events,and g(dz₁,dz₂) is the bivariate normal distribution with correlationparameter ρ, and the notation otherwise corresponds to the notation usedin the description above of DBAR Range Derivatives.

The following information includes the indices, the trading periods, thepredetermined termination criteria, the total amount invested and thevalue units used in this Example 3.1.6:

-   -   Asset Class 1: JP Morgan United States Government Bond Index        (“JPMGBI”)    -   Asset Class 1 Forward Price at Observation: 250.0    -   Asset Class 1 Volatility: 5%    -   Asset Class 2: S&P 500 Equity Index (“SP500”)    -   Asset Class 2 Forward Price at Observation: 1410    -   Asset Class 2 Volatility: 18%    -   Correlation Between Asset Classes: 0.5    -   Contingent Claim Expiration: Dec. 31, 1999    -   Trading Start Date: Jun. 30, 1999    -   Current Trading Period Start Date: Jul. 30, 1999    -   Current Trading Period End Date: Jul. 30, 1999    -   Next Trading Period Start Date: Aug. 2, 1999    -   Next Trading Period End Date: Aug. 31, 1999    -   Current Date: Jul. 12, 1999    -   Last Trading Period End Date: Dec. 30, 1999    -   Aggregate Investment for Current Trading Period: $100 million    -   Trade and Payout Value Units: U.S. Dollars        Table 3.1.6 shows the illustrative distribution of state returns        over the defined states for the joint outcomes based on this        information, with the defined states as indicated.

TABLE 3.1.6-1 Unit Returns for Joint Performance of S&P 500 and JPMGBIJPMGBI (0, (233, (237, (241, (244, (246, (248, (250, (252, (255, (257,(259, (264, (268, State 233] 237] 241] 244] 246] 248] 250] 252] 255]257] 259] 264] 268] ∞]   (0, 1102] 246 240 197 413 475 591 798 1167 17883039 3520 2330 11764 18518 (1102, 1174] 240 167 110 197 205 230 281 373538 841 1428 1753 7999 11764 (1174, 1252] 197 110 61 99 94 98 110 135180 259 407 448 1753 5207 (1252, 1292] 413 197 99 145 130 128 136 157197 269 398 407 1428 5813 (1292, 1334] 475 205 94 130 113 106 108 120144 189 269 259 841 3184 (1334, 1377] 591 230 98 128 106 95 93 99 115144 197 180 538 1851 SP500 (1377, 1421] 798 281 110 136 108 93 88 89 99120 157 135 373 1167 (1421, 1467] 1167 373 135 157 120 99 89 88 93 108136 110 281 798 (1467, 1515] 1851 538 180 197 144 115 99 93 95 106 12898 230 591 (1515, 1564] 3184 841 259 269 189 144 120 108 106 113 130 94205 475 (1564, 1614] 5813 1428 407 398 269 197 157 136 128 130 145 99197 413 (1614, 1720] 5207 1753 448 407 259 180 135 110 98 94 99 61 110197 (1720, 1834] 11764 7999 1753 1428 841 538 373 281 230 205 197 110167 240 (1834, ∞] 18518 11764 2330 3520 3039 1788 1167 798 591 475 413197 240 246

In Table 3.1.6-1, each cell contains the unit returns to the joint statereflected by the row and column entries. For example, the unit return toinvestments in the state encompassing the joint occurrence of the JPMGBIclosing on expiration at 249 and the SP500 closing at 1380 is 88. Sincethe correlation between two indices in this example is assumed to be0.5, the probability both indices will change in the same direction isgreater that the probability that both indices will change in oppositedirections. In other words, as represented in Table 3.1.6-1, unitreturns to investments in states represented in cells in the upper leftand lower right of the table—i.e., where the indices are changing in thesame direction—are lower, reflecting higher implied probabilities, thanunit returns to investments to states represented in cells in the lowerleft and upper right of Table 3.1.6-1 —i.e., where the indices arechanging in opposite directions.

As in the previous examples and in preferred embodiments, the returnsillustrated in Table 3.1.6-1 could be calculated as opening indicativereturns at the start of each trading period based on an estimate of whatthe closing returns for the trading period are likely to be. Theseindicative or opening returns can serve as an “anchor point” forcommencement of trading in a group of DBAR contingent claims. Of course,actual trading and trader expectations may induce substantial departuresfrom these indicative values.

Demand-based markets or auctions can be structured to trade DBARcontingent claims, including, for example, digital options, based onmultiple underlying events or variables and their inter-relationships.Market participants often have views about the joint outcome of twounderlying events or assets. Asset allocation managers, for example, areconcerned with the relative performance of bonds versus equities. Anadditional example of multivariate underlying events follows:

-   -   Joint Performance: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on the joint performance or        observation of two different variables. For example, digital        options traded in a demand-based market or auction can be based        on an underlying event defined as the joint observation of        non-farm payrolls and the unemployment rate.

Example 3.1.7 Corporate Bond Credit Risk

Groups of DBAR contingent claims can also be constructed on creditevents, such as the event that one of the major credit rating agencies(e.g., Standard and Poor's, Moodys) changes the rating for some or allof a corporation's outstanding securities. Indicative returns at theoutset of trading for a group of DBAR contingent claims oriented to acredit event can readily be constructed from publicly available datafrom the rating agencies themselves. For example, Table 3.1.7-1 containsindicative returns for an assumed group of DBAR contingent claims basedon the event that a corporation's Standard and Poor's credit rating fora given security will change over a certain period of time. In thisexample, states are defined using the Standard and Poor's creditcategories, ranging from AAA to D (default). Using the methods of thepresent invention, the indicative returns are calculated usinghistorical data on the frequency of the occurrence of these definedstates. In this example, a transaction fee of 1% is charged against theaggregate amount invested in the group of DBAR contingent claims, whichis assumed to be $100 million.

TABLE 3.1.7-1 Illustrative Returns for Credit DBAR Contingent Claimswith 1% Transaction Fee Current To New Historical Invested in StateIndicative Return to Rating Rating Probability ($) State A− AAA 0.0016160,000 617.75 A− AA+ 0.0004 40,000 2474.00 A− AA 0.0012 120,000 824.00A− AA− 0.003099 309,900 318.46 A− A+ 0.010897 1,089,700 89.85 A− A0.087574 8,757,400 10.30 A− A− 0.772868 77,286,800 0.28 A− BBB+ 0.0689796,897,900 13.35 A− BBB 0.03199 3,199,000 29.95 A− BBB− 0.007398 739,800132.82 A− BB+ 0.002299 229,900 429.62 A− BB 0.004999 499,900 197.04 A−BB− 0.002299 229,900 429.62 A− B+ 0.002699 269,900 365.80 A− B 0.000440,000 2474.00 A− B− 0.0004 40,000 2474.00 A− CCC 1E−04 10,000 9899.00A− D 0.0008 80,000 1236.50

In Table 3.1.7-1, the historical probabilities over the mutuallyexclusive and collectively exhaustive states sum to unity. Asdemonstrated above in this specification, in preferred embodiments, thetransaction fee affects the probability implied for each state from theunit return for that state.

Actual trading is expected almost always to alter illustrativeindicative returns based on historical empirical data. This Example3.1.7 indicates how efficiently groups of DBAR contingent claims can beconstructed for all traders or firms exposed to particular credit riskin order to hedge that risk. For example, in this Example, if a traderhas significant exposure to the A− rated bond issue described above, thetrader could want to hedge the event corresponding to a downgrade byStandard and Poor's. For example, this trader may be particularlyconcerned about a downgrade corresponding to an issuer default or “D”rating. The empirical probabilities suggest a payout of approximately$1,237 for each dollar invested in that state. If this trader has$100,000,000 of the corporate issue in his portfolio and a recovery ofratio of 0.3 can be expected in the event of default, then, in order tohedge $70,000,000 of default risk, the trader might invest in the stateencompassing a “D” outcome. To hedge the entire amount of the defaultrisk in this example, the amount of the investment in this state shouldbe $70,000,000/$1,237 or $56,589. This represents approximately 5.66basis points of the trader's position size in this bond (i.e.,$56,589/$100,000,000=0.00056)] which probably represents a reasonablecost of credit insurance against default. Actual investments in thisgroup of DBAR contingent claims could alter the return on the “D” eventover time and additional insurance might need to be purchased.

Demand-based markets or auctions can be structured to offer a widevariety of products related to common measures of credit quality,including Moody's and S&P ratings, bankruptcy statistics, and recoveryrates. For example, DBAR contingent claims can be based on an underlyingevent defined as the credit quality of Ford corporate debt as defined bythe Standard & Poor's rating agency.

Example 3.1.8 Economic Statistics

As financial markets have become more sophisticated, statisticalinformation that measures economic activity has assumed increasingimportance as a factor in the investment decisions of marketparticipants. Such economic activity measurements may include, forexample, the following U.S. federal government and U.S. and foreignprivate agency statistics:

-   -   Employment, National Output, and Income (Non-farm Payrolls,        Gross Domestic Product, Personal Income)    -   Orders, Production, and Inventories (Durable Goods Orders,        Industrial Production, Manufacturing Inventories)    -   Retail Sales, Housing Starts, Existing Home Sales, Current        Account Balance, Employment Cost Index, Consumer Price Index,        Federal Funds Target Rate    -   Agricultural statistics released by the U.S.D.A. (crop reports,        etc.)    -   The National Association of Purchasing Management (NAPM) survey        of manufacturing    -   Standard and Poor's Quarterly Operating Earnings of the S&P 500    -   The semiconductor book-to-bill ratio published by the        Semiconductor Industry Association    -   The Halifax House Price Index used extensively as an        authoritative indicator of house price movements in the U.K.        Because the economy is the primary driver of asset performance,        every investor that takes a position in equities, foreign        exchange, or fixed income will have exposure to economic forces        driving these asset prices, either by accident or design.        Accordingly, market participants expend considerable time and        resources to assemble data, models and forecasts. In turn,        corporations, governments, and financial intermediaries depend        heavily on the economic forecasts to allocate resources and to        make market projections.

To the extent that economic forecasts are inaccurate, inefficiencies andsevere misallocation of resources can result. Unfortunately, traditionalderivatives markets fail to provide market participants with a directmechanism to protect themselves against the adverse consequences offalling demand or rising input prices on a macroeconomic level.Demand-based markets or auctions for economic products, however, providemarket participants with a market price for the risk that a particularmeasure of economic activity will vary from expectations and a tool toproperly hedge the risk. The market participants can trade in a marketor an auction where the event of economic significance is an underlyingmeasure of economic activity (e.g., the VIX index as calculated by theCBOE) or a measured parameter related to the underlying event (e.g., animplied volatility or standard deviation of the VIX index).

For example, traders often hedge inflation risk by trading in bondfutures or, where they exist, inflation-protected floating rate bonds. Agroup of DBAR contingent claims can readily be constructed to allowtraders to express expectations about the distribution of uncertaineconomic statistics measuring, for example, the rate of inflation orother relevant variables. The following information describes such agroup of claims:

-   -   Economic Statistic: United States Non-Farm Payrolls    -   Announcement Date: May 31, 1999    -   Last Announcement Date: Apr. 30, 1999    -   Expiration: Announcement Date, May 31, 1999    -   Trading Start Date: May 1, 1999    -   Current Trading Period Start Date: May 10, 1999    -   Current Trading Period End Date: May 14, 1999    -   Current Date May 11, 1999    -   Last Announcement: 128,156 ('000)    -   Source: Bureau of Labor Statistics    -   Consensus Estimate: 130,000 (+1.2%)    -   Aggregate Amount Invested in Current Period: $100 million    -   Transaction Fee: 2.0% of Aggregate Traded amount

Using methods and systems of the present invention, states can bedefined and indicative returns can be constructed from, for example,consensus estimates among economists for this index. These estimates canbe expressed in absolute values or, as illustrated, in Table 3.1.8-1 inpercentage changes from the last observation as follows:

TABLE 3.1.8-1 Illustrative Returns For Non-Farm Payrolls Release with 2%Transaction Fee % Chg. In Index Investment in State Implied State State(′000) State Returns Probability [−100, −5] 100 979 0.001 (−5, −3] 200489 0.002 (−3, −1] 400 244 0.004 (−1, −.5] 500 195 0.005 (−.5, 0] 100097 0.01 (0, .5] 2000 48 0.02 (.5, .7] 3000 31.66667 0.03 (.7, .8] 400023.5 0.04 (.8, .9] 5000 18.6 0.05 (.9, 1.0] 10000 8.8 0.1 (1.0, 1.1]14000 6 0.14 (1.1, 1.2] 22000 3.454545 0.22 (1.2, 1.25] 18000 4.4444440.18 (1.25, 1.3] 9000 9.888889 0.09 (1.3, 1.35] 6000 15.33333 0.06(1.35, 1.40] 3000 31.66667 0.03 (1.40, 1.45] 200 489 0.002 (1.45, 1.5]600 162.3333 0.006 (1.5, 1.6] 400 244 0.004 (1.6, 1.7] 100 979 0.001(1.7, 1.8] 80 1224 0.0008 (1.8, 1.9] 59 1660.017 0.00059 (1.9, 2.0] 591660.017 0.00059 (2.0, 2.1] 59 1660.017 0.00059 (2.1, 2.2] 59 1660.0170.00059 (2.2, 2.4] 59 1660.017 0.00059 (2.4, 2.6] 59 1660.017 0.00059(2.6, 3.0] 59 1660.017 0.00059 (3.0, ∞] 7 13999 0.00007As in examples, actual trading prior to the trading end date would beexpected to adjust returns according to the amounts invested in eachstate and the total amount invested for all the states.

Demand-based markets or auctions can be structured to offer a widevariety of products related to commonly observed indices and statisticsrelated to economic activity and released or published by governments,and by domestic, foreign and international government or privatecompanies, institutions, agencies or other entities. These may include alarge number of statistics that measure the performance of the economy,such as employment, national income, inventories, consumer spending,etc., in addition to measures of real property and other economicactivity. An additional example follows:

-   -   Private Economic Indices & Statistics: Demand-based markets or        auctions can be structured to trade DBAR contingent claims,        including, for example, digital options, based on economic        statistics released or published by private sources. For        example, DBAR contingent claims can be based on an underlying        event defined as the NAPM Index published by the National        Association of Purchasing Managers.        -   Alternative private indices might also include measures of            real property. For example, DBAR contingent claims,            including, for example, digital options, can be based on an            underlying event defined as the level of the Halifax House            Price Index at year-end, 2001.    -   In addition to the general advantages of the demand-based        trading system, demand-based products on economic statistics        will provide the following new opportunities for trading and        risk management:

-   (1) Insuring against the event risk component of asset price    movements. Statistical releases can often cause extreme short-term    price movements in the fixed income and equity markets. Many market    participants have strong views on particular economic reports, and    try to capitalize on such views by taking positions in the bond or    equity markets. Demand-based markets or auctions on economic    statistics provide participants with a means of taking a direct view    on economic variables, rather than the indirect approach employed    currently.

-   (2) Risk management for real economic activity. State governments,    municipalities, insurance companies, and corporations may all have a    strong interest in a particular measure of real economic activity.    For example, the Department of Energy publishes the Electric Power    Monthly which provides electricity statistics at the State, Census    division, and U.S. levels for net generation, fossil fuel    consumption and stocks, quantity and quality of fossil fuels, cost    of fossil fuels, electricity retail sales, associated revenue, and    average revenue. Demand-based markets or auctions based on one or    more of these energy benchmarks can serve as invaluable risk    management mechanisms for corporations and governments seeking to    manage the increasingly uncertain outlook for electric power.

-   (3) Sector-specific risk management. The Health Care CPI (Consumer    Price Index) published by the U.S. Bureau of Labor Statistics tracks    the CPI of medical care on a monthly basis in the CPI Detailed    Report. A demand-based market or auction on this statistic would    have broad applicability for insurance companies; drug companies,    hospitals, and many other participants in the health care industry.    Similarly, the semiconductor book-to-bill ratio serves as a direct    measure of activity in the semiconductor equipment manufacturing    industry. The ratio reports both shipments and new bookings with a    short time lag, and hence is a useful measure of supply and demand    balance in the semiconductor industry. Not only would manufacturers    and consumers of semiconductors have a direct financial interest,    but the ratio's status as a bellwether of the general technology    market would invite participation from financial market participants    as well.

Example 3.1.9 Corporate Events

Corporate actions and announcements are further examples of events ofeconomic significance which are usually unhedgable or uninsurable intraditional markets but which can be effectively structured into groupsof DBAR contingent claims according to the present invention.

In recent years, corporate earnings expectations, which are typicallyannounced on a quarterly basis for publicly traded companies, haveassumed increasing importance as more companies forego dividends toreinvest in continuing operations. Without dividends, the present valueof an equity becomes entirely dependent on revenues and earnings streamsthat extend well into the future, causing the equity itself to take onthe characteristics of an option. As expectations of future cash flowschange, the impact on pricing can be dramatic, causing stock prices inmany cases to exhibit option-like behavior.

Traditionally, market participants expend considerable time andresources to assemble data, models and forecasts. To the extent thatforecasts are inaccurate, inefficiencies and severe misallocation ofresources can result. Unfortunately, traditional derivatives marketsfail to provide market participants with a direct mechanism to managethe unsystematic risks of equity ownership. Demand-based markets orauctions for corporate earnings and revenues, however, provide marketparticipants with a concrete price for the risk that earnings andrevenues may vary from expectations and permit them to insure or hedgeor speculate on the risk.

Many data services, such as IBES and FirstCall, currently publishestimates by analysts and a consensus estimate in advance of quarterlyearnings announcements. Such estimates can form the basis for indicativeopening returns at the commencement of trading in a demand-based marketor auction as illustrated below. For this example, a transaction fee ofzero is assumed.

-   -   Underlying security: IBM    -   Earnings Announcement Date: Jul. 21, 1999    -   Consensus Estimate: 0.879/share    -   Expiration: Announcement, Jul. 21, 1999    -   First Trading Period Start Date: Apr. 19, 1999    -   First Trading Period End Date May 19, 1999    -   Current Trading Period Start Date: Jul. 6, 1999    -   Current Trading Period End Date: Jul. 9, 1999    -   Next Trading Period Start Date: Jul. 9, 1999    -   Next Trading Period End Date: Jul. 16, 1999    -   Total Amount Invested in Current Trading Period: $100 million

TABLE 3.1.9-1 Illustrative Returns For IBM Earnings AnnouncementEarnings Invested in State State0 (′000 $) Unit Returns Implied StateProbability (−∞, .5] 70 1,427.57 0.0007 (.5, .6] 360 276.78 0.0036 (.6,.65] 730 135.99 0.0073 (.65, .7] 1450 67.97 0.0145 (.7, .74] 2180 44.870.0218 (.74, .78] 3630 26.55 0.0363 (.78, ..8] 4360 21.94 0.0436 (.8,.82] 5820 16.18 0.0582 (.82, .84] 7270 12.76 0.0727 (.84, .86] 872010.47 0.0872 (.86, .87] 10900 8.17 0.109 (.87, .88] 18170 4.50 0.1817(.88, .89] 8720 10.47 0.0872 (.89, .9] 7270 12.76 0.0727 (.9, .91] 509018.65 0.0509 (.91, .92] 3630 26.55 0.0363 (.92, .93] 2910 33.36 0.0291(.93, .95] 2180 44.87 0.0218 (.95, .97] 1450 67.97 0.0145 (.97, .99]1310 75.34 0.0131 (.99, 1.1] 1160 85.21 0.0116 (1.1, 1.3] 1020 97.040.0102 (1.3, 1.5] 730 135.99 0.0073 (1.5, 1.7] 360 276.78 0.0036 (1.7,1.9] 220 453.55 0.0022 (1.9, 2.1] 150 665.67 0.0015 (2.1, 2.3] 701,427.57 0.0007 (2.3, 2.5] 40 2,499.00 0.0004 (2.5, ∞] 30 3,332.330.0003Consistent with the consensus estimate, the state with the largestinvestment encompasses the range (0.87, 0.88].

TABLE 3.1.9-2 Illustrative Returns for Microsoft Earnings AnnouncementStrike Bid Offer Payout Volume Calls <40 0.9525 0.9575 1.0471 4,100,000<41 0.9025 0.9075 1.1050 1,000,000 <42 0.8373 0.8423 1.1908 9,700 <430.7475 0.7525 1.3333 3,596,700 <44 0.622 0.627 1.6013 2,000,000 <450.4975 0.5025 2.0000 6,000,000 <46 0.3675 0.3725 2.7027 2,500,000 <470.2175 0.2225 4.5455 1,000,000 <48 0.1245 0.1295 7.8740 800,000 <490.086 0.091 11.2994 — <50 0.0475 0.0525 20.000 194,700 Puts <40 0.04250.0475 22.2222 193,100 <41 0.0925 0.0975 10.5263 105,500 <42 0.15770.1627 6.2422 — <43 0.2475 0.2525 4.0000 1,200,000 <44 0.3730 0.37802.6631 1,202,500 <45 0.4975 0.5025 2.0000 6,000,000 <46 0.6275 0.63251.5873 4,256,600 <47 0.7775 0.7825 1.2821 3,545,700 <48 0.8705 0.87551.1455 5,500,000 <49 0.9090 0.9140 1.0971 — <50 0.9475 0.9525 1.05263,700,000

The table above provides a sample distribution of trades that might bemade for an April 23 auction period for Microsoft Q4 corporate earnings(June 2001), due to be released on Jul. 16, 2001.

For example, at 29 times trailing earnings and 28 times consensus 2002earnings, Microsoft is experiencing single digit profit growth and isthe object of uncertainty with respect to sales of Microsoft Office,adoption rates of Windows 2000, and the .Net initiative. In the sampledemand-based market or auction based on earnings expectations depictedabove, a market participant can engage, for example, in the followingtrading tactics and strategies with respect to DBAR digital options.

-   -   A fund manager wishing to avoid market risk at the current time        but who still wants exposure to Microsoft can buy the 0.43        Earnings per Share Call (consensus currently 0.44-45) with        reasonable confidence that reported earnings will be 43 cents or        higher. Should Microsoft report earnings as expected, the trader        earns approximately 33% on invested demand-based trading digital        option premium (i.e., 1/option price of 0.7525). Conversely,        should Microsoft report earnings below 43 cents, the invested        premium would be lost, but the consequences for Microsoft's        stock price would likely be dramatic.    -   A more aggressive strategy would involve selling or        underweighting Microsoft stock, while purchasing a string of        digital options on higher than expected EPS growth. In this        case, the trader expects a multiple contraction to occur over        the short to medium term, as the valuation becomes        unsustainable. Using the market for DBAR contingent claims on        earnings depicted above, a trader with a $5 million notional        exposure to Microsoft can buy a string of digital call options,        as follows:

Strike Premium Price Net Payout .46 $37,000 0.3725 $62,329 .47 22,0000.2225 139,205 .48 6,350 0.1295 181,890 .49 4,425 0.0910 226,091 .50 00.0525 226,091

-   -   -   The payouts displayed immediately above are net of premium            investment. Premiums invested are based on the trader's            assessment of likely stock price (and price multiple)            reaction to a possible earnings surprise. Similar trades in            digital options on earnings would be made in successive            quarters, resulting in a string of options on higher than            expected earnings growth, to protect against an upward shift            in the earnings expectation curve, as shown in FIG. 21.        -   The total cost, for this quarter, amounts to $69,775, just            above a single quarter's interest income on the notional            $5,000,000, invested at 5%.

    -   A trader with a view on a range of earnings expectations for the        quarter can profit from a spread strategy over the distribution.        By purchasing the 0.42 call and selling the 0.46 call, the        trader can construct a digital option spread priced at:        0.8423-0.3675=0.4748. This spread would, consequently, pay out:        1/0.4748=2.106, for every dollar invested.

Many trades can be constructed using demand-based trading for DBARcontingent claims, including, for example, digital options, based oncorporate earnings. The examples shown here are intended to berepresentative, not definitive. Moreover, demand-based trading productscan be based on corporate accounting measures, including a wide varietyof generally accepted accounting information from corporate balancesheets, income statements, and other measures of cash flow, such asearnings before interest, taxes, depreciation, and amortization(EBITDA). The following examples provide a further representativesampling:

-   -   Revenues: Demand-based markets or auctions for DBAR contingent        claims, including, for example, digital options can be based on        a measure or parameter related to Cisco revenues, such as the        gross revenues reported by the Cisco Corporation. The underlying        event for these claims is the quarterly or annual gross revenue        figure for Cisco as calculated and released to the public by the        reporting company.    -   EBITDA (Earnings Before Interest, Taxes, Depreciation,        Amortization): Demand-based markets or auctions for DBAR        contingent claims, including, for example, digital options can        be based on a measure or parameter related to AOL EBITDA, such        as the EBITDA figure reported by AOL that is used to provide a        measure of operating earnings. The underlying event for these        claims is the quarterly or annual EBITDA figure for AOL as        calculated and released to the public by the reporting company.        In addition to the general advantages of the demand-based        trading system, products based on corporate earnings and        revenues may provide the following new opportunities for trading        and risk management:

-   (1) Trading the price of a stock relative to its earnings. Traders    can use a market for earnings to create a “Multiple Trade,” in which    a stock would be sold (or ‘not owned’) and a string of DBAR    contingent claims, including, for example, digital options, based on    quarterly earnings can be used as a hedge or insurance for stock    believed to be overpriced. Market expectations for a company's    earnings may be faulty, and may threaten the stability of a stock    price, post announcement. Corporate announcements that reduce    expectation for earnings and earnings growth highlight the    consequences for high-multiple growth stocks that fail to meet    expectations. For example, an equity investment manager might decide    to underweight a high-multiple stock against a benchmark, and    replace it with a series of DBAR digital options corresponding to a    projected profile for earnings growth. The manager can compare the    cost of this strategy with the risk of owning the underlying    security, based on the company's PE ratio or some other metric    chosen by the fund manager. Conversely, an investor who expects a    multiple expansion for a given stock would purchase demand-based    trading digital put options on earnings, retaining the stock for a    multiple expansion while protecting against a shortfall in reported    earnings.

-   (2) Insuring against an earnings shortfall, while maintaining a    stock position during a period when equity options are deemed too    expensive. While DBAR contingent claims, including, for example,    digital options, based on earnings are not designed to hedge stock    prices, they can provide a cost-effective means to mitigate the risk    of equity ownership over longer term horizons. For example,    periodically, three-month stock options that are slightly    out-of-the-money can command premiums of 10% or more. The ability to    insure against possible earnings or revenue shortfalls one quarter    or more in the future via purchases of DBAR digital options may    represent an attractive alternative to conventional hedge strategies    for equity price risks.

-   (3) Insuring against an earnings shortfall that may trigger credit    downgrades. Fixed income managers worried about potential exposure    to credit downgrades from reduced corporate earnings can use DBAR    contingent claims, including, for example, digital options, to    protect against earnings shortfalls that would impact EBITDA and    prompt declines in corporate bond prices. Conventional fixed income    and convertible bond managers can protect against equity exposures    without a short sale of the corresponding equity shares.

-   (4) Obtaining low-risk, incremental returns. Market participants can    use deep-in-the-money DBAR contingent claims, including, for    example, digital options, based on earnings as a source of low-risk,    uncorrelated returns.

Example 3.1.10 Real Assets

Another advantage of the methods and systems of the present invention isthe ability to structure liquid claims on illiquid underlying assetssuch a real estate. As previously discussed, traditional derivativesmarkets customarily use a liquid underlying market in order to functionproperly. With a group of DBAR contingent claims all that is usuallyrequired is a real-world, observable event of economic significance. Forexample, the creation of contingent claims tied to real assets has beenattempted at some financial institutions over the last several years.These efforts have not been credited with an appreciable impact,apparently because of the primary liquidity constraints inherent in theunderlying real assets.

A group of DBAR contingent claims according to the present invention canbe constructed based on an observable event related to real estate. Therelevant information for an illustrative group of such claims is asfollows:

-   -   Real Asset Index: Colliers ABR Manhattan Office Rent Rates    -   Bloomberg Ticker: COLAMANR    -   Update Frequency: Monthly    -   Source: Colliers ABR, Inc.    -   Announcement Date: Jul. 31, 1999    -   Last Announcement Date: Jun. 30, 1999    -   Last Index Value: $45.39/sq. ft.    -   Consensus Estimate: $45.50    -   Expiration: Announcement Jul. 31, 1999    -   Current Trading Period Start: Jun. 30, 1999    -   Current Trading Period End: Jul. 7, 1999    -   Next Trading Period Start: Jul. 7, 1999    -   Next Trading Period End: Jul. 14, 1999

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be calculated or willemerge from actual trader investments according to the methods of thepresent invention as illustrated in Examples 3.1.1-3.1.9.

Demand-based markets or auctions can be structured to offer a widevariety of products related to real assets, such as real estate,bandwidth, wireless spectrum capacity, or computer memory. An additionalexample follows:

-   -   Computer Memory: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on computer memory components.        For example, DBAR contingent claims can be based on an        underlying event defined as the 64 Mb (8×8) PC 133 DRAM memory        chip prices and on the rolling 90-day average of Dynamic Random        Access Memory DRAM prices as reported each Friday by ICIS-LOR, a        commodity price monitoring group based in London.

Example 3.1.11 Energy Supply Chain

A group of DBAR contingent claims can also be constructed using themethods and systems of the present invention to provide hedging vehicleson non-tradable quantities of great economic significance within thesupply chain of a given industry. An example of such an application isthe number of oil rigs currently deployed in domestic U.S. oilproduction. The rig count tends to be a slowly adjusting quantity thatis sensitive to energy prices. Thus, appropriately structured groups ofDBAR contingent claims based on rig counts could enable suppliers,producers and drillers to hedge exposure to sudden changes in energyprices and could provide a valuable risk-sharing device.

For example, a group of DBAR contingent claims depending on the rigcount could be constructed according to the present invention using thefollowing information (e.g., data source, termination criteria, etc).

-   -   Asset Index: Baker Hughes Rig Count U.S. Total    -   Bloomberg Ticker: BAKETOT    -   Frequency: Weekly    -   Source: Baker Hughes, Inc.    -   Announcement Date: Jul. 16, 1999    -   Last Announcement Date: Jul. 9, 1999    -   Expiration Date: Jul. 16, 1999    -   Trading Start Date: Jul. 9, 1999    -   Trading End Date: Jul. 15, 1999    -   Last: 570    -   Consensus Estimate: 580

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9. A varietyof embodiments of DBAR contingent claims, including for example, digitaloptions, can be based on an underlying event defined as the Baker HughesRig Count observed on a semi-annual basis.

-   -   Demand-based markets or auctions can be structured to offer a        wide variety of products related to power and emissions,        including electricity prices, loads, degree-days, water supply,        and pollution credits. The following examples provide a further        representative sampling:    -   Electricity Prices: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on the price of electricity at        various points on the electricity grid. For example, DBAR        contingent claims can be based on an underlying event defined as        the weekly average price of electricity in kilowatt-hours at the        New York Independent System Operator (NYISO).    -   Transmission Load Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on the actual load (power        demand) experienced for a particular power pool, allowing        participants to trade volume, in addition to price. For example,        DBAR contingent claims can be based on an underlying event        defined as the weekly total load demand experienced by        Pennsylvania-New Jersey-Maryland Interconnect (PJM Western Hub).    -   Water: Demand-based markets or auctions can be structured to        trade DBAR contingent claims, including, for example, digital        options, based on water supply. Water measures are useful to a        broad variety of constituents, including power companies,        agricultural producers, and municipalities. For example, DBAR        contingent claims can be based on an underlying event defined as        the cumulative precipitation observed at weather stations        maintained by the National Weather Service in the Northwest        catchment area, including Washington, Idaho, Montana, and        Wyoming.    -   Emission Allowances Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on emission allowances for        various pollutants. For example, DBAR contingent claims can be        based on an underlying event defined as price of Environmental        Protection Agency (EPA) sulfur dioxide allowances at the annual        market or auction administered by the Chicago Board of Trade.

Example 3.1.12 Mortgage Prepayment Risk

Real estate mortgages comprise an extremely large fixed income assetclass with hundreds of billions in market capitalization. Marketparticipants generally understand that these mortgage-backed securitiesare subject to interest rate risk and the risk that borrowers mayexercise their options to refinance their mortgages or otherwise“prepay” their existing mortgage loans. The owner of a mortgagesecurity, therefore, bears the risk of being “called” out of itsposition when mortgage interest rate levels decline.

Market participants expend considerable time and resources assemblingeconometric models and synthesizing various data populations in order togenerate prepayment projections. To the extent that economic forecastsare inaccurate, inefficiencies and severe misallocation of resources canresult. Unfortunately, traditional derivatives markets fail to providemarket participants with a direct mechanism to protect themselvesagainst a homeowner's exercise of its prepayment option. Demand-basedmarkets or auctions for mortgage prepayment products, however, providemarket participants with a concrete price for prepayment risk.

Groups of DBAR contingent claims can be structured according to thepresent invention, for example, based on the following information:

-   -   Asset Index: FNMA Conventional 30 year One-Month Historical        Aggregate Prepayments    -   Coupon: 6.5%    -   Frequency: Monthly    -   Source: Bloomberg    -   Announcement Date: Aug. 1, 1999    -   Last Announcement Date: Jul. 1, 1999    -   Expiration: Announcement Date, Aug. 1, 1999    -   Current Trading Period Start Date: Jul. 1, 1999    -   Current Trading Period End Date: Jul. 9, 1999    -   Last: 303 Public Securities Association Prepayment Speed (“PSA”)    -   Consensus Estimate: 310 PSA

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

In addition to the general advantages of the demand-based tradingsystem, products on mortgage prepayments may provide the followingexemplary new opportunities for trading and risk management:

-   (1) Asset-specific applications. In the simplest form, the owner of    a prepayable mortgage-backed security carries, by definition, a    series of short option positions embedded in the asset, whereas a    DBAR contingent claim, including, for example, a digital option,    based on mortgage prepayments would constitute a long option    position. A security owner would have the opportunity to compare the    digital option's expected return with the prospective loss of    principal, correlate the offsetting options, and invest accordingly.    While this tactic would not eliminate reinvestment risks, per se, it    would generate incremental investment returns that would reduce the    security owner's embedded liabilities with respect to short option    positions.-   (2) Portfolio applications. Certainly, a similar strategy could be    applied on an expanded basis to a portfolio of mortgage-backed    securities, or a portfolio of whole mortgage loans.-   (3) Enhancements to specific pools. Certain pools of seasoned    mortgage loans exhibit consistent prepayment patterns, based upon    comprehensible factors—origination period, underwriting standards,    borrower circumstances, geographic phenomena, etc. Because of    homogeneous prepayment performance, mortgage market participants can    obtain greater confidence with respect to the accuracy of    predictions for prepayments in these pools, than in the case of    pools of heterogeneous, newly originated loans that lack a    prepayment history. Market conventions tend to assign lower    volatility estimates to the correlation of prepayment changes in    seasoned pools for given interest rate changes, than in the case of    newer pools. A relatively consistent prepayment pattern for seasoned    mortgage loan pools would heighten the certainty of correctly    anticipating future prepayments, which would heighten the likelihood    of consistent success in trading in DBAR contingent claims such as,    for example, digital options, based on respective mortgage    prepayments. Such digital option investments, combined with seasoned    pools, would tend to enhance annuity-like cash profiles, and reduce    investment risks.-   (4) Prepayment puts plus discount MBS. Discount mortgage-backed    securities tend to enjoy two-fold benefits as interest rates decline    in the form of positive price changes and increases in prepayment    speeds. Converse penalties apply in events of increases in interest    rates, where a discount MBS suffers from adverse price change, and a    decline in prepayment income. A discount MBS owner could offset    diminished prepayment income by investing in DBAR contingent claims,    such as, for example, digital put options, or digital put option    spreads on prepayments. An analogous strategy would apply to    principal-only mortgage-backed securities.-   (5) Prepayment calls plus premium MBS. An expectation of interest    rate declines that accelerate prepayment activity for premium    mortgage-backed securities would motivate a premium bond-holder to    purchase DBAR contingent claims, such as, for example, digital call    options, based on mortgage prepayments to offset losses attributable    to unwelcome paydowns. The analogue would also apply to    interest-only mortgage-backed securities.-   (6) Convexity additions. An investment in a DBAR contingent claim,    such as, for example, a digital option, based on mortgage    prepayments should effectively add convexity to an interest rate    sensitive investment. According to this reasoning, dollar-weighted    purchases of a demand-based market or auction on mortgage    prepayments would tend to offset the negative convexity exhibited by    mortgage-backed securities. It is likely that expert participants in    the mortgage marketplace will analyze and test, and ultimately    harvest, the fruitful opportunities for combinations of DBAR    contingent claims, including, for example, digital options, based on    mortgage prepayments with mortgage-backed securities and    derivatives.

Example 3.1.13 Insurance Industry Loss Warranty (“ILW”)

The cumulative impact of catastrophic and non-catastrophic insurancelosses over the past two years has reduced the capital available in theretrocession market (i.e. reinsurance for reinsurance companies) andpushed up insurance and reinsurance rates for property catastrophecoverage. Because large reinsurance companies operate global businesseswith global exposures, severe losses from catastrophes in one countrytend to drive up insurance and reinsurance rates for unrelated perils inother countries simply due to capital constraints.

As capital becomes scarce and insurance rates increase, marketparticipants usually access the capital markets by purchasingcatastrophic bonds (CAT bonds) issued by special purpose reinsurancecompanies. The capital markets can absorb the risk of loss associatedwith larger disasters, whereas a single insurer or even a group ofinsurers cannot, because the risk is spread across many more marketparticipants.

Unlike traditional capital markets that generally exhibit a naturaltwo-way order flow, insurance markets typically exhibit one-way demandgenerated by participants desiring protection from adverse outcomes.Because demand-based trading products do not require an underlyingsource of supply, such products provide an attractive alternative foraccess to capital.

Groups of DBAR contingent claims can be structured using the system andmethods of the present invention to provide insurance and reinsurancefacilities for property and casualty, life, health and other traditionallines of insurance. The following information provides information tostructure a group of DBAR contingent claims related to large propertylosses from hurricane damage:

-   -   Event: PCS Eastern Excess $5 billion Index    -   Source: Property Claim Services (PCS)    -   Frequency: Monthly    -   Announcement Date: Oct. 1, 1999    -   Last Announcement Date: Jul. 1, 1999    -   Last Index Value: No events    -   Consensus Estimate: $1 billion (claims excess of $5 billion)    -   Expiration: Announcement Date, Oct. 1, 1999    -   Trading Period Start Date: Jul. 1, 1999    -   Trading Period End Date: Sep. 30, 1999

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

In preferred embodiments of groups of DBAR contingent claims related toproperty-casualty catastrophe losses, the frequency of claims and thedistributions of the severity of losses are assumed and convolutions areperformed in order to post indicative returns over the distribution ofdefined states. This can be done, for example, using compoundfrequency-severity models, such as the Poisson-Pareto model, familiar tothose of skill in the art, which predict, with greater probability thana normal distribution, when losses will be extreme. As indicatedpreviously, in preferred embodiments market activity is expected toalter the posted indicative returns, which serve as informative levelsat the commencement of trading.

Demand-based markets or auctions can be structured to offer a widevariety of products related to insurance industry loss warranties andother insurable risks, including property and non-property catastrophe,mortality rates, mass torts, etc. An additional example follows:

-   -   Property Catastrophe: Demand-based markets or auctions can be        based on the outcome of natural catastrophes, including        earthquake, fire, atmospheric peril, and flooding, etc.        Underlying events can be based on hazard parameters. For        example, DBAR contingent claims can be based on an underlying        event defined as the cumulative losses sustained in California        as the result of earthquake damage in the year 2002, as        calculated by the Property Claims Service (PCS).

In addition to the general advantages of the demand-based tradingsystem, products on catastrophe risk will provide the following newopportunities for trading and risk management:

-   (1) Greater transaction efficiency and precision. A demand-based    trading catastrophe risk product, such as, for example, a DBAR    digital option, allows participants to buy or sell a precise    notional quantity of desired risk, at any point along a catastrophe    risk probability curve, with a limit price for the risk. A series of    loss triggers can be created for catastrophic events that offer    greater flexibility and customization for insurance transactions, in    addition to indicative pricing for all trigger levels. Segments of    risk coverage can be traded with ease and precision. Participants in    demand-based trading catastrophe risk products gain the ability to    adjust risk protection or exposure to a desired level. For example,    a reinsurance company may wish to purchase protection at the tail of    a distribution, for unlikely but extremely catastrophic losses,    while writing insurance in other parts of the distribution where    returns may appear attractive.-   (2) Credit quality. Claims-paying ability of an insurer or reinsurer    represents an important concern for many market participants.    Participants in a demand-based market or auction do not depend on    the credit quality of an individual insurance or reinsurance    company. A demand-based market or auction is by nature self-funding,    meaning that catastrophic losses in other product or geographic    areas will not impair the ability of a demand-based trading    catastrophe risk product to make capital distributions.

Example 3.1.14 Conditional Events

As discussed above, advantage of the systems and methods of the presentinvention is the ability to construct groups of DBAR contingent claimsrelated to events of economic significance for which there is greatinterest in insurance and hedging, but which are not readily hedged orinsured in traditional capital and insurance markets. Another example ofsuch an event is one that occurs only when some related event haspreviously occurred. For purposes of illustration, these two events maybe denoted A and B.

${q{\langle{AB}\rangle}} = \frac{q\left( {A\bigcap B} \right)}{q(B)}$

where q denotes the probability of a state, q

A|B

represents the conditional probability of state A given the prioroccurrence of state and B, and q(A∩B) represents the occurrence of bothstates A and B.

For example, a group of DBAR contingent claims may be constructed tocombine elements of “key person” insurance and the performance of thestock price of the company managed by the key person. Many firms aremanaged by people whom capital markets perceive as indispensable orparticularly important, such as Warren Buffett of Berkshire Hathaway.The holders of Berkshire Hathaway stock have no ready way of insuringagainst the sudden change in management of Berkshire, either due to acorporate action such as a takeover or to the death or disability ofWarren Buffett. A group of conditional DBAR contingent claims can beconstructed according to the present invention where the defined statesreflect the stock price of Berkshire Hathaway conditional on WarrenBuffet's leaving the firm's management. Other conditional DBARcontingent claims that could attract significant amounts for investmentcan be constructed using the methods and systems of the presentinvention, as apparent to one of skill in the art.

Example 3.1.15 Securitization Using a DBAR Contingent Claim Mechanism

The systems and methods of the present invention can also be adapted bya financial intermediary or issuer for the issuance of securities suchas bonds, common or preferred stock, or other types of financialinstruments. The process of creating new opportunities for hedgingunderlying events through the creation of new securities is known as“securitization,” and is also discussed in an embodiment presented inSection 10. Well-known examples of securitization include the mortgageand asset-backed securities markets, in which portfolios of financialrisk are aggregated and then recombined into new sources of financialrisk. The systems and methods of the present invention can be usedwithin the securitization process by creating securities, or portfoliosof securities, whose risk, in whole or part, is tied to an associated orembedded group of DBAR contingent claims. In a preferred embodiment, agroup of DBAR contingent claims is associated with a security much likeoptions are currently associated with bonds in order to create callableand putable bonds in the traditional markets.

This example illustrates how a group of DBAR contingent claims accordingto the present invention can be tied to the issuance of a security inorder to share risk associated with an identified future event among thesecurity holders. In this example, the security is a fixed income bondwith an embedded group of DBAR contingent claims whose value depends onthe possible values for hurricane losses over some time period for somegeographic region.

-   -   Issuer: Tokyo Fire and Marine    -   Underwriter: Goldman Sachs    -   DBAR Event: Total Losses on a Saffir-Simpson Category 4        Hurricane    -   Geographic: Property Claims Services Eastern North America    -   Date: Jul. 1, 1999-Nov. 1, 1999    -   Size of Issue: 500 million USD.    -   Issue Date: Jun. 1, 1999    -   DBAR Trading Period: Jun. 1, 1999-Jul. 1, 1999

In this example, the underwriter Goldman Sachs issues the bond, andholders of the issued bond put bond principal at risk over the entiredistribution of amounts of Category 4 losses for the event. Ranges ofpossible losses comprise the defined states for the embedded group ofDBAR contingent claims. In a preferred embodiment, the underwriter isresponsible for updating the returns to investments in the variousstates, monitoring credit risk, and clearing and settling, andvalidating the amount of the losses. When the event is determined anduncertainty is resolved, Goldman is “put” or collects the bond principalat risk from the unsuccessful investments and allocates these amounts tothe successful investments. The mechanism in this illustration thusincludes:

-   -   (1) An underwriter or intermediary which implements the        mechanism, and    -   (2) A group of DBAR contingent claims directly tied to a        security or issue (such as the catastrophe bond above).

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

Example 3.1.16 Exotic Derivatives

The securities and derivatives communities frequently use the term“exotic derivatives” to refer to derivatives whose values are linked toa security, asset, financial product or source of financial risk in amore complicated fashion than traditional derivatives such as futures,call options, and convertible bonds. Examples of exotic derivativesinclude American options, Asian options, barrier options, Bermudanoptions, chooser and compound options, binary or digital options,lookback options, automatic and flexible caps and floors, and shoutoptions.

Many types of exotic options are currently traded. For example, barrieroptions are rights to purchase an underlying financial product, such asa quantity of foreign currency, for a specified rate or price, but onlyif, for example, the underlying exchange rate crosses or does not crossone or more defined rates or “barriers.” For example, a dollar call/yenput on the dollar/yen exchange rate, expiring in three months withstrike price 110 and “knock-out” barrier of 105, entitles the holder topurchase a quantity of dollars at 110 yen per dollar, but only if theexchange rate did not fall below 105 at any point during the three monthduration of the option. Another example of a commonly traded exoticderivative, an Asian option, depends on the average value of theunderlying security over some time period. Thus, a class of exoticderivatives is commonly referred to as “path-dependent” derivatives,such as barrier and Asian options, since their values depend not only onthe value of the underlying financial product at a given date, but on ahistory of the value or state of the underlying financial product.

The properties and features of exotic derivatives are often so complexso as to present a significant source of “model risk” or the risk thatthe tools, or the assumptions upon which they are based, will lead tosignificant errors in pricing and hedging. Accordingly, derivativestraders and risk managers often employ sophisticated analytical tools totrade, hedge, and manage the risk of exotic derivatives.

One of the advantages of the systems and methods of the presentinvention is the ability to construct groups of DBAR contingent claimswith exotic features that are more manageable and transparent thantraditional exotic derivatives. For example, a trader might beinterested in the earliest time the yen/dollar exchange rate crosses 95over the next three months. A traditional barrier option, or portfolioof such exotic options, might suffice to approximate the source of riskof interest to this trader. A group of DBAR contingent claims, incontrast, can be constructed to isolate this risk and present relativelytransparent opportunities for hedging. A risk to be isolated is thedistribution of possible outcomes for what barrier derivatives tradersterm the “first passage time,” or, in this example, the first time thatthe yen/dollar exchange rate crosses 95 over the next three months.

The following illustration shows how such a group of DBAR contingentclaims can be constructed to address this risk. In this example, it isassumed that all traders in the group of claims agree that theunderlying exchange rate is lognormally distributed. This group ofclaims illustrates how traders would invest in states and thus expressopinions regarding whether and when the forward yen/dollar exchange ratewill cross a given barrier over the next 3 months:

-   -   Underlying Risk: Japanese/U.S. Dollar Yen Exchange Rate    -   Current Date: Sep. 15, 1999    -   Expiration: Forward Rate First Passage Time, as defined, between        Sep. 16, 1999 to Dec. 16, 1999    -   Trading Start Date: Sep. 15, 1999    -   Trading End Date: Sep. 16, 1999    -   Barrier: 95    -   Spot JPY/USD: 104.68    -   Forward JPY/USD 103.268    -   Assumed (Illustrative) Market Volatility: 20% annualized    -   Aggregate Traded Amount: 10 million USD

TABLE 3.1.16-1 First Passage Time for Yen/Dollar Dec. 16, 1999 ForwardExchange Rate Return Time in Year Fractions Invested in State (′000) PerUnit if State Occurs (0, .005] 229.7379 42.52786 (.005, .01] 848.902410.77992 (.01, .015] 813.8007 11.28802 (.015, .02] 663.2165 14.07803(.02, .025] 536.3282 17.6453 (.025 .03] 440.5172 21.70059 (.03, .035]368.4647 26.13964 (.035, .04] 313.3813 30.91 (.04, .045] 270.420735.97942 (.045, .05] 236.2651 41.32534 (.05, .075] 850.2595 10.76112(.075, .1] 540.0654 17.51627 (.1, .125] 381.3604 25.22191 (.125, .15]287.6032 33.77013 (.15, .175] 226.8385 43.08423 (.175, .2] 184.823853.10558 (.2, .225] 154.3511 63.78734 (.225, .25] 131.4217 75.09094 DidNot Hit Barrier 2522.242 2.964727

As with other examples, and in preferred embodiments, actual tradingwill likely generate traded amounts and therefore returns that departfrom the assumptions used to compute the illustrative returns for eachstate.

In addition to the straightforward multivariate events outlined above,demand-based markets or auctions can be used to create and trade digitaloptions (as described in Sections 6 and 7) on calculated underlyingevents (including the events described in this Section 3), similar tothose found in exotic derivatives. Many exotic derivatives are based onpath-dependent outcomes such as the average of an underlying event overtime, price thresholds, a multiple of the underlying, or some sort oftime constraint. An additional example follows:

-   -   Path Dependent: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, on an underlying event that is the        subject of a calculation. For example, digital options traded in        a demand-based market or auction could be based on an underlying        event defined as the average price of yen/dollar exchange rate        for the last quarter of 2001.

Example 3.1.17 Hedging Markets for Real Goods, Commodities and Services

Investment and capital budgeting choices faced by firms typicallyinvolve inherent economic risk (e.g., future demand for semiconductors),large capital investments (e.g., semiconductor fabrication capacity) andtiming (e.g., a decision to invest in a plant now, or defer for someperiod of time). Many economists who study such decisions underuncertainty have recognized that such choices involve what they term“real options.”This characterization indicates that the choice to investnow or to defer an investment in goods or services or a plant, forexample, in the face of changing uncertainty and information, frequentlyentails risks similar to those encountered by traders who have investedin options which provide the opportunity to buy or sell an underlyingasset in the capital markets. Many economists and investors recognizethe importance of real options in capital budgeting decisions and ofsetting up markets to better manage their uncertainty and value. Naturalresource and extractive industries, such as petroleum exploration andproduction, as well as industries requiring large capital investmentssuch as technology manufacturing, are prime examples of industries wherereal options analysis is increasingly used and valued.

Groups of DBAR contingent claims according to the present invention canbe used by firms within a given industry to better analyze capitalbudgeting decisions, including those involving real options. Forexample, a group of DBAR contingent claims can be established whichprovides hedging opportunities over the distribution of futuresemiconductor prices. Such a group of claims would allow producers ofsemiconductors to better hedge their capital budgeting decisions andprovide information as to the market's expectation of future prices overthe entire distribution of possible price outcomes. This informationabout the market's expectation of future prices could then also be usedin the real options context in order to better evaluate capitalbudgeting decisions. Similarly, computer manufacturers could use suchgroups of DBAR contingent claims to hedge against adverse semiconductorprice changes.

Information providing the basis for constructing an illustrative groupof DBAR contingent claims on semiconductor prices is as follows:

-   -   Underlying Event: Semiconductor Monthly Sales    -   Index: Semiconductor Industry Association Monthly Global Sales        Release    -   Current Date: Sep. 15, 1999    -   Last Release Date: Sep. 2, 1999    -   Last Release Month: July, 1999    -   Last Release Value: 11.55 Billion, USD    -   Next Release Date: Approx. Oct. 1, 1999    -   Next Release Month: August 1999    -   Trading Start Date: Sep. 2, 1999    -   Trading End Date: Sep. 30, 1999

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in previous examples.

Groups of DBAR contingent claims according to the present invention canalso be used to hedge arbitrary sources of risk due to price discoveryprocesses. For example, firms involved in competitive bidding for goodsor services, whether by sealed bid or open bid markets or auctions, canhedge their investments and other capital expended in preparing the bidby investing in states of a group of DBAR contingent claims comprisingranges of mutually exclusive and collectively exhaustive market orauction bids. In this way, the group of DBAR contingent claim serves asa kind of “meta-auction,” and allows those who will be participating inthe market or auction to invest in the distribution of possible marketor auction outcomes, rather than simply waiting for the single outcomerepresenting the market or auction result. Market or auctionparticipants could thus hedge themselves against adverse market orauction developments and outcomes, and, importantly, have access to theentire probability distribution of bids (at least at one point in time)before submitting a bid into the real market or auction. Thus, a groupof DBAR claims could be used to provide market data over the entiredistribution of possible bids. Preferred embodiments of the presentinvention thus can help avoid the so-called Winner's Curse phenomenonknown to economists, whereby market or auction participants failrationally to take account of the information on the likely bids oftheir market or auction competitors.

Demand-based markets or auctions can be structured to offer a widevariety of products related to commodities such as fuels, chemicals,base metals, precious metals, agricultural products, etc. The followingexamples provide a further representative sampling:

-   -   Fuels: Demand-based markets or auctions can be based on measures        related to various fuel sources. For example, DBAR contingent        claims, including, e.g., digital options, can be based on an        underlying event defined as the price of natural gas in Btu's        delivered to the Henry Hub, Louisiana.    -   Chemicals: Demand-based markets or auctions can be based on        measures related to a variety of other chemicals. For example,        DBAR contingent claims, including, e.g., digital options, can be        based on an underlying event defined as the price of        polyethylene.    -   Base Metals: Demand-based markets or auctions can be based on        measures related to various precious metals. For example, DBAR        contingent claims, including, e.g., digital options, can be        based on an underlying event defined as the price per gross ton        of #1 Heavy Melt Scrap Iron.    -   Precious Metals: Demand-based markets or auctions can be based        on measures related to various precious metals. For example,        DBAR contingent claims, including, e.g., digital options, can be        based on an underlying event defined as the price per troy ounce        of Platinum delivered to an approved storage facility.    -   Agricultural Products: Demand-based markets or auctions can be        based on measures related to various agricultural products. For        example, DBAR contingent claims, including, e.g., digital        options, can be based on an underlying event defined as the        price per bushel of #2 yellow corn delivered at the Chicago        Switching District.

Example 3.1.18 DBAR Hedging

Another feature of the systems and methods of the present invention isthe relative ease with which traders can hedge risky exposures. In thefollowing example, it is assumed that a group of DBAR contingent claimshas two states (state 1 and state 2, or s₁ or s₂), and amounts T₁, andT₂ are invested in state 1 and state 2, respectively. The unit payout π₁for state 1 is therefore T₂/T₁ and for state 2 it is T₁/T₂. If a traderthen invests amount α₁ in state 1, and state 1 then occurs, the traderin this example would receive the following payouts, P, indexed by theappropriate state subscripts:

$P_{1} = {\alpha_{1}*\left( {\frac{T_{2}}{T_{1} + \alpha_{1}} + 1} \right)}$

If state 2 occurs the trader would receive

P ₂=0

If, at some point during the trading period, the trader desires to hedgehis exposure, the investment in state 2 to do so is calculated asfollows:

$\alpha_{2} = {\frac{\alpha_{1}*T_{2}}{T_{1}}.}$

This is found by equating the state payouts with the proposed hedgetrade, as follows:

$P_{1} = {{\alpha_{1}*\left( {\frac{T_{2} + \alpha_{2}}{T_{1} + \alpha_{1}} + 1} \right)} = {P_{2} = {\alpha_{2}*\left( {\frac{T_{1} + \alpha_{1}}{T_{2} + \alpha_{2}} + 1} \right)}}}$

Compared to the calculation required to hedge traditional derivatives,these expressions show that, in appropriate groups of DBAR contingentclaims of the present invention, calculating and implementing hedges canbe relatively straightforward.

The hedge ratio, α₂, just computed for a simple two state example can beadapted to a group of DBAR contingent claims which is defined over morethan two states. In a preferred embodiment of a group of DBAR contingentclaims, the existing investments in states to be hedged can bedistinguished from the states on which a future hedge investment is tobe made. The latter states can be called the “complement” states, sincethey comprise all the states that can occur other than those in whichinvestment by a trader has already been made, i.e., they arecomplementary to the invested states. A multi-state hedge in a preferredembodiment includes two steps: (1) determining the amount of the hedgeinvestment in the complement states, and (2) given the amount sodetermined, allocating the amount among the complement states. Theamount of the hedge investment in the complement states pursuant to thefirst step is calculated as:

$\alpha_{C} = \frac{\alpha_{H}*T_{C}}{T_{H}}$

where α_(C) is amount of the hedge investment in the complement states,α_(H) is the amount of the existing investment in the states to behedged, T_(c) is the existing amount invested in the complement states,and T_(H) is the amount invested the states to be hedged, exclusive ofα_(H). The second step involves allocating the hedge investment amongthe complement states, which can be done by allocating α_(C) among thecomplement states in proportion to the existing amounts already investedin each of those states.

An example of a four-state group of DBAR contingent claims according tothe present invention illustrates this two-step hedging process. Forpurposes of this example, the following assumptions are made: (i) thereare four states, numbered 1 through 4, respectively; (ii) $50, $80, $70and $40 is invested in each state, (iii) a trader has previously placeda multi-state investment in the amount of $10 (α_(H) as defined above)for states 1 and 2; and (iv) the allocation of this multi-stateinvestment in states 1 and 2 is $3.8462 and $6.15385, respectively. Theamounts invested in each state, excluding the trader's invested amounts,are therefore $46.1538, $73.84615, $70, and $40 for states 1 through 4,respectively. It is noted that the amount invested in the states to behedged, i.e., states 1 and 2, exclusive of the multi-state investment of$10, is the quantity T_(H) as defined above.

The first step in a preferred embodiment of the two-step hedging processis to compute the amount of the hedge investment to be made in thecomplement states. As derived above, the amount of the new hedgeinvestment is equal to the amount of the existing investment multipliedby the ratio of the amount invested in the complement states to theamount invested in the states to be hedged, excluding the trader'sexisting trades, i.e., $10*($70+$40)/($46.1538+$73.84615)=$9.16667. Thesecond step in this process is to allocate this amount between the twocomplement states, i.e., states 3 and 4.

Following the procedures discussed above for allocating multi-stateinvestments, the complement state allocation is accomplished byallocating the hedge investment amount—$9.16667 in this example—inproportion to the existing amount previously invested in the complementstates, i.e., $9.16667*$70/$110=$5.83333 for state 3 and$9.16667*$40/$110=$3.3333 for state 4. Thus, in this example, the tradernow has the following amounts invested in states 1 through 4: ($3.8462,$6.15385, $5.8333, $3.3333); the total amount invested in each of thefour states is $50, $80, $75.83333, and $43.3333); and the returns foreach of the four states, based on the total amount invested in each ofthe four states, would be, respectively, (3.98333, 2.1146, 2.2857, and4.75). In this example, if state 1 occurs the trader will receive apayout, including the amount invested in state 1, of3.98333*$3.8462+$3.8462=$19.1667 which is equal to the sum invested, sothe trader is fully hedged against the occurrence of state 1.Calculations for the other states yield the same results, so that thetrader in this example would be fully hedged irrespective of which stateoccurs.

As returns can be expected to change throughout the trading period, thetrader would correspondingly need to rebalance both the amount of hishedge investment for the complement states as well as the multi-stateallocation among the complement states. In a preferred embodiment, aDBAR contingent claim exchange can be responsible for reallocatingmulti-state trades via a suspense account, for example, so the tradercan assign the duty of reallocating the multi-state investment to theexchange. Similarly, the trader can also assign to an exchange theresponsibility of determining the amount of the hedge investment in thecomplement states especially as returns change as a result of trading.The calculation and allocation of this amount can be done by theexchange in a similar fashion to the way the exchange reallocatesmulti-state trades to constituent states as investment amounts change.

Example 3.1.19 Quasi-Continuous Trading

Preferred embodiments of the systems and methods of the presentinvention include a trading period during which returns adjust amongdefined states for a group of DBAR contingent claims, and a laterobservation period during which the outcome is ascertained for the eventon which the group of claims is based. In preferred embodiments, returnsare allocated to the occurrence of a state based on the finaldistribution of amounts invested over all the states at the end of thetrading period. Thus, in each embodiments a trader will not know hisreturns to a given state with certainty until the end of a given tradingperiod. The changes in returns or “price discovery” which occur duringthe trading period prior to “locking-in” the final returns may provideuseful information as to trader expectations regarding finalizedoutcomes, even though they are only indications as to what the finalreturns are going to be. Thus, in some preferred embodiments, a tradermay not be able to realize profits or losses during the trading period.The hedging illustration of Example 3.1.18, for instance, provides anexample of risk reduction but not of locking-in or realizing profit andloss.

In other preferred embodiments, a quasi-continuous market for trading ina group of DBAR contingent claims may be created. In preferredembodiments, a plurality of recurring trading periods may providetraders with nearly continuous opportunities to realize profit and loss.In one such embodiment, the end of one trading period is immediatelyfollowed by the opening of a new trading period, and the final investedamount and state returns for a prior trading period are “locked in” asthat period ends, and are allocated accordingly when the outcome of therelevant event is later known. As a new trading period begins on thegroup of DBAR contingent claims related to the same underlying event, anew distribution of invested amounts for states can emerge along with acorresponding new distribution of state returns. In such embodiments, asthe successive trading periods are made to open and close morefrequently, a quasi-continuous market can be obtained, enabling tradersto hedge and realize profit and loss as frequently as they currently doin the traditional markets.

An example illustrates how this feature of the present invention may beimplemented. The example illustrates the hedging of a European digitalcall option on the yen/dollar exchange rate (a traditional marketoption) over a two day period during which the underlying exchange ratechanges by one yen per dollar. In this example, two trading periods areassumed for the group of DBAR contingent claims

Traditional Option: European Digital Option

Payout of Option: Pays 100 million USD if exchange rate equals orexceeds strike price at maturity or expirationUnderlying Index Yen/dollar exchange rate

Option Start: Aug. 12, 1999 Option Expiration Aug. 15, 2000

Assumed Volatility: 20% annualized

Strike Price: 120

Notional: 100 million USD

In this example, two dates are analyzed, Aug. 12, 1999 and Aug. 13,1999:

TABLE 3.1.19-1 Change in Traditional Digital Call Option Value Over TwoDays Observation Date Aug. 12, 1999 Aug. 13, 1999 Spot Settlement DateAug. 16, 1999 Aug. 17, 1999 Spot Price for Settlement Date 115.55 116.55Forward Settlement Date Aug. 15, 2000 Aug. 15, 2000 Forward Price109.217107 110.1779 Option Premium 28.333% of Notional 29.8137% ofNotional

Table 3.1.19-1 shows how the digital call option struck at 120 could, asan example, change in value with an underlying change in the yen/dollarexchange rate. The second column shows that the option is worth 28.333%or $28.333 million on a $100 million notional on Aug. 12, 1999 when theunderlying exchange rate is 115.55. The third column shows that thevalue of the option, which pays $100 million should dollar yen equal orexceed 120 at the expiration date, increases to 29.8137% or $29.8137million per $100 million when the underlying exchange rate has increasedby 1 yen to 116.55. Thus, the traditional digital call option generatesa profit of $29.81377−$28.333=$1.48077 million.

This example shows how this profit also could be realized in trading ina group of DBAR contingent claims with two successive trading periods.It is also assumed for purposes of this example that there aresufficient amounts invested, or liquidity, in both states such that theparticular trader's investment does not materially affect the returns toeach state. This is a convenient but not necessary assumption thatallows the trader to take the returns to each state “as given” withoutconcern as to how his investment will affect the closing returns for agiven trading period. Using information from Table 3.1.19-1, thefollowing closing returns for each state can be derived:

Trading Period 1:

-   -   Current trading period end date: Aug. 12, 1999    -   Underlying Event: Closing level of yen/dollar exchange rate for        Aug. 15, 2000 settlement, 4 pm EDT    -   Spot Price for Aug. 16, 1999 Settlement: 115.55

State JPY/USD < 120 for JPY/USD ≧ 120 Aug. 15, 2000 for Aug. 15, 2000Closing Returns 0.39533 2.5295

For purposes of this example, it is assumed that an illustrative traderhas $28.333 million invested in the state that the yen/dollar exchangerate equals or exceeds 120 for Aug. 15, 2000 settlement.

Trading Period 2:

-   -   Current trading period end date: Aug. 13, 1999    -   Underlying Event: Closing level of dollar/yen exchange rate for        Aug. 15, 2000 settlement, 4 pm EDT    -   Spot Price for Aug. 17, 1999 Settlement: 116.55

State JPY/USD < 120 JPY/USD ≧ 120 for Aug. 15, 2000 for Aug. 15, 2000Closing State Returns .424773 2.3542

For purposes of this example, it is also assumed that the illustrativetrader has a $70.18755 million hedging investment in the state that theyen/dollar exchange rate is less than 120 for Aug. 15, 2000 settlement.It is noted that, for the second period, the closing returns are lowerfor the state that the exchange equals or exceeds 120. This is due tothe change represented in Table 3.1.19-1 reflecting an assumed change inthe underlying market, which would make that state more likely.

The trader now has an investment in each trading period and has lockedin a profit of $1.4807 million, as shown below:

State JPY/USD < 120 JPY/USD ≧ 120 for Aug. 15, 2000 for Aug. 15, 2000Profit and Loss $70.18755 * .424773 − $−70.18755 + 28.333 * (000.000)$28.333 = $1.48077 $2.5295 = $1.48077

The illustrative trader in this example has therefore been able tolock-in or realize the profit no matter which state finally occurs. Thisprofit is identical to the profit realized in the traditional digitaloption, illustrating that systems and methods of the present inventioncan be used to provide at least daily if not more frequent realizationof profits and losses, or that risks can be hedged in virtually realtime.

In preferred embodiments, a quasi-continuous time hedge can beaccomplished, in general, by the following hedge investment, assumingthe effect of the size of the hedge trade does not materially effect thereturns:

$H = {\alpha_{t}*\frac{1 + r_{t}}{1 + r_{t + 1}^{c}}}$

-   -   where r_(t)=closing returns a state in which an investment was        originally made at time t    -   α_(t)=amount originally invested in the state at time t    -   r^(c) _(t+1)=closing returns at time t+1 to state or states        other than the state in which the original investment was made        (i.e., the so-called complement states which are all states        other than the state or states originally traded which are to be        hedged)    -   H=the amount of the hedge investment

If H is to be invested in more than one state, then a multi-stateallocation among the constituent states can be performed using themethods and procedures described above. This expression for H allowsinvestors in DBAR contingent claims to calculate the investment amountsfor hedging transactions. In the traditional markets, such calculationsare often complex and quite difficult.

Example 3.1.20 Value Units For Investments and Payouts

As previously discussed in this specification, the units of investmentsand payouts used in embodiments of the present invention can be any unitof economic value recognized by investors, including, for example,currencies, commodities, number of shares, quantities of indices,amounts of swap transactions, or amounts of real estate. The investedamounts and payouts need not be in the same units and can comprise agroup or combination of such units, for example 25% gold, 25% barrels ofoil, and 50% Japanese Yen. The previous examples in this specificationhave generally used U.S. dollars as the value units for investments andpayouts.

This Example 3.1.20 illustrates a group of DBAR contingent claims for acommon stock in which the invested units and payouts are defined inquantities of shares. For this example, the terms and conditions ofExample 3.1.1 are generally used for the group of contingent claims onMSFT common stock, except for purposes of brevity, only three states arepresented in this Example 3.1.20: (0,83], (83, 88], and (88,∞]. Also inthis Example 3.1.20, invested amounts are in numbers of shares for eachstate and the exchange makes the conversion for the trader at the marketprice prevailing at the time of the investment. In this example, payoutsare made according to a canonical DRF in which a trader receives aquantity of shares equal to the number of shares invested in states thatdid not occur, in proportion to the ratio of number of shares the traderhas invested in the state that did occur, divided by the total number ofshares invested in that state. An indicative distribution of traderdemand in units of number of shares is shown below, assuming that thetotal traded amount is 100,000 shares:

Return Per Share if State Occurs Amount Traded in Number Unit Returns inNumber of State of Share Shares  (0, 83] 17,803 4.617 (83, 88] 72,725.37504 (88, ∞]  9,472 9.5574

If, for instance, MSFT closes at 91 at expiration, then in this examplethe third state has occurred, and a trader who had previously invested10 shares in that state would receive a payout of 10*9.5574+10=105.574shares which includes the trader's original investment. Traders who hadpreviously invested in the other two states would lose all of theirshares upon application of the canonical DRF of this example.

An important feature of investing in value units other than units ofcurrency is that the magnitude of the observed outcome may well berelevant, as well as the state that occurs based on that outcome. Forexample, if the investments in this example were made in dollars, thetrader who has a dollar invested in state (88,∞] would not care, atleast in theory, whether the final price of MSFT at the close of theobservation period were 89 or 500. However, if the value units arenumbers of shares of stock, then the magnitude of the final outcome doesmatter, since the trader receives as a payout a number of shares whichcan be converted to more dollars at a higher outcome price of $91 pershare. For instance, for a payout of 105.574 shares, these shares areworth 105.574*$91=$9,607.23 at the outcome price. Had the outcome pricebeen $125, these shares would have been worth 105.574*125=$13,196.75.

A group of DBAR contingent claims using value units of commodity havinga price can therefore possess additional features compared to groups ofDBAR contingent claims that offer fixed payouts for a state, regardlessof the magnitude of the outcome within that state. These features mayprove useful in constructing groups of DBAR contingent claims which areable to readily provide risk and return profiles similar to thoseprovided by traditional derivatives. For example, the group of DBARcontingent claims described in this example could be of great interestto traders who transact in traditional derivatives known as“asset-or-nothing digital options” and “supershares options.”

Example 3.1.21 Replication of An Arbitrary Payout Distribution

An advantage of the systems and methods of the present invention isthat, in preferred embodiments, traders can generate an arbitrarydistribution of payouts across the distribution of defined states for agroup of DBAR contingent claims. The ability to generate a customizedpayout distribution may be important to traders, since they may desireto replicate contingent claims payouts that are commonly found intraditional markets, such as those corresponding to long positions instocks, short positions in bonds, short options positions in foreignexchange, and long option straddle positions, to cite just a fewexamples. In addition, preferred embodiments of the present inventionmay enable replicated distributions of payouts which can only begenerated with difficulty and expense in traditional markets, such asthe distribution of payouts for a long position in a stock that issubject to being “stopped out” by having a market-maker sell the stockwhen it reaches a certain price below the market price. Such stop-lossorders are notoriously difficult to execute in traditional markets, andtraders are frequently not guaranteed that the execution will occurexactly at the pre-specified price.

In preferred embodiments, and as discussed above, the generation andreplication of arbitrary payout distributions across a givendistribution of states for a group of DBAR contingent claims may beachieved through the use of multi-state investments. In suchembodiments, before making an investment, traders can specify a desiredpayout for each state or some of the states in a given distribution ofstates. These payouts form a distribution of desired payouts across thedistribution of states for the group of DBAR contingent claims. Inpreferred embodiments, the distribution of desired payouts may be storedby an exchange, which may also calculate, given an existing distributionof investments across the distribution of states, (1) the total amountrequired to be invested to achieve the desired payout distribution; (2)the states into which the investment is to allocated; and (3) how muchis to be invested in each state so that the desired payout distributioncan be achieved. In preferred embodiments, this multi-state investmentis entered into a suspense account maintained by the exchange, whichreallocates the investment among the states as the amounts investedchange across the distribution of states. In preferred embodiments, asdiscussed above, a final allocation is made at the end of the tradingperiod when returns are finalized.

The discussion in this specification of multi-state investments hasincluded examples in which it has been assumed that an illustrativetrader desires a payout which is the same no matter which state occursamong the constituent states of a multi-state investment. To achievethis result, in preferred embodiments the amount invested by the traderin the multi-state investment can be allocated to the constituent statein proportion to the amounts that have otherwise been invested in therespective constituent states. In preferred embodiments, theseinvestments are reallocated using the same procedure throughout thetrading period as the relative proportion of amounts invested in theconstituent states changes.

In other preferred embodiments, a trader may make a multi-stateinvestment in which the multi-state allocation is not intended togenerate the same payout irrespective of which state among theconstituent state occurs. Rather, in such embodiments, the multi-stateinvestment may be intended to generate a payout distribution whichmatches some other desired payout distribution of the trader across thedistribution of states, such as, for example, for certain digitalstrips, as discussed in Section 6. Thus, the systems and methods of thepresent invention do not require amounts invested in multi-stateinvestments to be allocated in proportion of the amounts otherwiseinvested in the constituent states of the multi-statement investment.

Notation previously developed in this specification is used to describea preferred embodiment of a method by which replication of an arbitrarydistribution of payouts can be achieved for a group of DBAR contingentclaims according to the present invention. The following additionalnotation, is also used:

-   -   A_(i,*) denotes the i-th row of the matrix A containing the        invested amounts by trader i for each of the n states of the        group of DBAR contingent claims        In preferred embodiments, the allocation of amounts invested in        all the states which achieves the desired payouts across the        distribution of states can be calculated using, for example, the        computer code listing in Table 1 (or functional equivalents        known to one of skill in the art), or, in the case where a        trader's multi-state investment is small relative to the total        investments already made in the group of DBAR contingent claims,        the following approximation:

A _(i,*) ^(T)=Π⁻¹ *P _(i,*) ^(T)

where the −1 superscript on the matrix Π denotes a matrix inverseoperation. Thus, in these embodiments, amounts to be invested to producean arbitrary distribution payouts can approximately be found bymultiplying (a) the inverse of a diagonal matrix with the unit payoutsfor each state on the diagonal (where the unit payouts are determinedfrom the amounts invested at any given time in the trading period) and(b) a vector containing the trader's desired payouts. The equation aboveshows that the amounts to be invested in order to produce a desiredpayout distribution are a function of the desired payout distributionitself (P_(i,*)) and the amounts otherwise invested across thedistribution of states (which are used to form the matrix Π, whichcontains the payouts per unit along its diagonals and zeroes along theoff-diagonals). Therefore, in preferred embodiments, the allocation ofthe amounts to be invested in each state will change if either thedesired payouts change or if the amounts otherwise invested across thedistribution change. As the amounts otherwise invested in various statescan be expected to change during the course of a trading period, inpreferred embodiments a suspense account is used to reallocate theinvested amounts, A_(i,*), in response to these changes, as describedpreviously. In preferred embodiments, at the end of the trading period afinal allocation is made using the amounts otherwise invested across thedistribution of states. The final allocation can typically be performedusing the iterative quadratic solution techniques embodied in thecomputer code listing in Table 1.

Example 3.1.21 illustrates a methodology for generating an arbitrarypayout distribution, using the event, termination criteria, the definedstates, trading period and other relevant information, as appropriate,from Example 3.1.1, and assuming that the desired multi-state investmentis small in relation to the total amount of investments already made. InExample 3.1.1 above, illustrative investments are shown across thedistribution of states representing possible closing prices for MSFTstock on the expiration date of Aug. 19, 1999. In that example, thedistribution of investment is illustrated for Aug. 18, 1999, one dayprior to expiration, and the price of MSFT on this date is given as 85.For purposes of this Example 3.1.21, it is assumed that a trader wouldlike to invest in a group of DBAR contingent claims according to thepresent invention in a way that approximately replicates the profits andlosses that would result from owning one share of MSFT (i.e., arelatively small amount) between the prices of 80 and 90. In otherwords, it is assumed that the trader would like to replicate atraditional long position in MSFT with the restrictions that a sellorder is to be executed when MSFT reaches 80 or 90. Thus, for example,if MSFT closes at 87 on Aug. 19, 1999 the trader would expect to have $2of profit from appropriate investments in a group of DBAR contingentclaims. Using the defined states identified in Example 3.1.1, thisprofit would be approximate since the states are defined to include arange of discrete possible closing prices.

In preferred embodiments, an investment in a state receives the samereturn regardless of the actual outcome within the state. It istherefore assumed for purposes of this Example 3.1.21 that a traderwould accept an appropriate replication of the traditional profit andloss from a traditional position, subject to only “discretization”error. For purposes of this Example 3.1.21, and in preferredembodiments, it is assumed that the profit and loss corresponding to anactual outcome within a state is determined with reference to the pricewhich falls exactly in between the upper and lower bounds of the stateas measured in units of probability, i.e., the “state average.” For thisExample 3.1.21, the following desired payouts can be calculated for eachof the states the amounts to be invested in each state and the resultinginvestment amounts to achieve those payouts:

TABLE 3.1.21-1 Investment Which Generates Desired States State Average($) Desired Payout ($) Payout ($)  (0, 80] NA 80 0.837258 (80, 80.5]80.33673 80.33673 0.699493 (80.5, 81] 80.83349 80.83349 1.14091 (81,81.5] 81.33029 81.33029 1.755077 (81.5, 82] 81.82712 81.82712 2.549131(82, 82.5] 82.32401 82.32401 3.498683 (82.5, 83] 82.82094 82.820944.543112 (83, 83.5] 83.31792 83.31792 5.588056 (83.5, 84] 83.8149683.81496 6.512429 (84, 84.5] 84.31204 84.31204 7.206157 (84.5, 85]84.80918 84.80918 7.572248 (85, 85.5] 85.30638 85.30638 7.555924 (85.5,86] 85.80363 85.80363 7.18022 (86, 86.5] 86.30094 86.30094 6.493675(86.5, 87] 86.7983 86.7983 5.59628 (87, 87.5] 87.29572 87.29572 4.599353(87.5, 88] 87.7932 87.7932 3.611403 (88, 88.5] 88.29074 88.290742.706645 (88.5, 89] 88.78834 88.78834 1.939457 (89, 89.5] 89.2859989.28599 1.330046 (89.5, 90] 89.7837 89.7837 0.873212 (90, ∞] NA 901.2795The far right column of Table 3.1.21-1 is the result of the matrixcomputation described above. The payouts used to construct the matrix Πfor this Example 3.1.21 are one plus the returns shown in Example 3.1.1for each state.

Pertinently the systems and methods of the present invention may be usedto achieve almost any arbitrary payout or return profile, e.g., a longposition, a short position, an option “straddle”, etc., whilemaintaining limited liability and the other benefits of the inventiondescribed in this specification.

As discussed above, if many traders make multi-state investments, in apreferred embodiment an iterative procedure is used to allocate all ofthe multi-state investments to their respective constituent states.Computer code, as previously described and apparent to one of skill inthe art, can be implemented to allocate each multi-state investmentamong the constituent states depending upon the distribution of amountsotherwise invested and the trader's desired payout distribution.

Example 3.1.22 Emerging Market Currencies

Corporate and investment portfolio managers recognize the utility ofoptions to hedge exposures to foreign exchange movements. In the G7currencies, liquid spot and forward markets support an extremelyefficient options market. In contrast, many emerging market currencieslack the liquidity to support efficient, liquid spot and forward marketsbecause of their small economic base. Without ready access to a sourceof tradable underlying supply, pricing and risk control of options inemerging market currencies are difficult or impossible.

Governmental intervention and credit constraints further inhibittransaction flows in emerging market currencies. Certain governmentschoose to restrict the convertibility of their currency for a variety ofreasons, thus reducing access to liquidity at any price and effectivelypreventing option market-makers from gaining access to a tradableunderlying supply. Mismatches between sources of local liquidity andcreditworthy counterparties further restrict access to a tradableunderlying supply. Regional banks that service local customers haveaccess to indigenous liquidity but poor credit ratings whilemultinational commercial and investment banks with superior creditratings have limited access to liquidity. Because credit considerationsprevent external market participants from taking on significantexposures to local counterparties, transaction choices are limited.

The foreign exchange market has responded to this lack of liquidity bymaking use of non-deliverable forwards (NDFs) which, by definition, donot require an exchange of underlying currency. Although NDFs have metwith some success, their utility is still constrained by a lack ofliquidity. Moreover, the limited liquidity available to NDFs isgenerally insufficient to support an active options market.

Groups of DBAR contingent claims can be structured using the system andmethods of the present invention to support an active options market inemerging market currencies.

In addition to the general advantages of the demand-based tradingsystem, products on emerging market currencies will provide thefollowing new opportunities for trading and risk management:

-   (1) Credit enhancement. An investment bank can use demand-based    trading emerging market currency products to overcome existing    credit barriers. The ability of a demand-based market or auction to    process only buy orders, combined with the limited liability of    option payout profiles (vs. forward contracts), allows banks to    precisely define the limits of their counterparty credit exposure    and, hence, to trade with local market institutions, increasing    participation and liquidity.

Example 3.1.23 Central Bank Target Rates

Portfolio managers and market-makers formulate market views based inpart on their forecasts for future movements in central bank targetrates. When the Federal Reserve (Fed), European Central Bank (ECB) orBank of Japan (BOJ), for example, changes their target rate or whenmarket participants adjust their expectations about future rate moves,global equity and fixed income financial markets can react quickly anddramatically.

Market participants currently take views on central bank target rates bytrading 3-month interest rate futures, such as Eurodollar futures forthe Fed and Euribor futures for the ECB. Although these markets arequite liquid, significant risks impair trading in such contracts:futures contracts have a 3-month maturity while central bank targetrates change overnight; and models for credit spreads and term structureare required for futures pricing. Market participants additionallyexpress views on the target Fed funds rate by trading Fed funds futures,which are based on the overnight Fed funds rate. Although less riskythan Eurodollar futures, significant risks also impair trading in Fedfunds futures: the overnight Fed funds rate can differ, sometimessignificantly, from the target Fed funds rate due to overnight liquidityspikes and month-end effects; and, Fed funds futures frequently cannotaccommodate the full volumes that investment managers would like toexecute at a given market price.

Groups of DBAR contingent claims can be structured using the system andmethods of the present invention to develop an explicit mechanism bywhich market participants can express views regarding central banktarget rates. For example, demand-based markets or auctions can be basedon central bank policy parameters such as the Federal Reserve Target FedFunds Rate, the Bank of Japan Official Discount Rate, or the Bank ofEngland Base Rate. For example, the underlying event may be defined asthe Federal Reserve Target Fed Funds Rate as of Jun. 1, 2002. Becausedemand-based trading products settle using the target rate of interest,maturity and credit mismatches no longer pose market barriers.

In addition to the general advantages of the demand-based tradingsystem, products on central bank target rates may provide the followingnew advantages for trading and risk management:

-   (1) No basis risk. Since demand-based trading products settle using    the target rate of interest, there is no maturity mismatch and no    credit mismatch. Demand-based trading products for central bank    target rates have no basis risk.-   (2) An exact date match to central bank meetings. Demand-based    trading products can be structured to allow investors to take views    on specific meetings by matching the date of expiry of a contract    with the date of the central bank meeting.-   (3) A direct way to express views on intra-meeting moves.    Demand-based trading products allow special tailoring so that    portfolio managers can take a view on whether or not a central bank    will change its target rate intra-meeting.-   (4) Managing the event risk associated with a central bank meeting.    Almost all market participants have portfolios that are    significantly affected by shifts in target rates. Market    participants can use demand-based trading options on central bank    target rates to lower their portfolio's overall volatility.-   (5) Managing short-term funding costs. Banks and large corporations    often borrow short-term funds at a rate highly correlated with    central bank target rates, e.g., U.S. banks borrow at a rate that    closely follows target Fed funds. These institutions may better    manage their funding costs using demand-based trading products on    central bank rates.

Example 3.1.24 Weather

In recent years, market participants have expressed increasing interestin a market for derivative instruments related to weather as a means toinsure against adverse weather outcomes. Despite greater recognition ofthe role of weather in economic activity, the market for weatherderivatives has been relatively slow to develop. Market-makers intraditional over-the-counter markets often lack the means toredistribute their risk because of limited liquidity and lack of anunderlying instrument. The market for weather derivatives is furtherhampered by poor price discovery.

A group of DBAR contingent claims can be constructed using the methodsand systems of the present invention to provide market participants witha market price for the probability that a particular weather metric willbe above or below a given level. For example, participants in ademand-based market or auction on cooling degree days (CDDs) or onheating degree days (HDDs) in New York from Nov. 1, 2001 through Mar.31, 2002 may be able to see at a glance the market consensus price thatcumulative CDDs or HDDs will exceed certain levels. The eventobservation could be specified as taking place at a preset location suchas the Weather Bureau Army Navy Observation Station #14732.Alternatively, participants in a demand-based market or auction onwind-speed in Chicago may be able to see at a glance the marketconsensus price that cumulative wind-speeds will exceed certain levels.

Example 3.1.25 Financial Instruments

Demand-based markets or auctions can be structured to offer a widevariety of products on commonly offered financial instruments orstructured financial products related to fixed income securities,equities, foreign exchange, interest rates, and indices, and anyderivatives thereof. When the underlying economic event is a change (ordegree of change) in a financial instrument or product, the possibleoutcomes can include changes which are positive, negative or equal tozero when there is no change, and amounts of each positive and negativechange. The following examples provide a further representativesampling:

-   -   Equity Prices: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on prices for equity securities        listed on recognized exchanges throughout the world. For        example, DBAR contingent claims can be based on an underlying        event defined as the closing price each week of Juniper        Networks. The underlying event can also be defined using an        alternative measure, such as the volume weighted average price        during any day.    -   Fixed Income Security Prices: Demand-based markets or auctions        can be structured to trade DBAR contingent claims, including,        for example, digital options, based on a variety of fixed income        securities such as government T-bills, T-notes, and T-bonds,        commercial paper, CD's, zero coupon bonds, corporate, and        municipal bonds, and mortgage-backed securities. For example,        DBAR contingent claims can be based on an underlying event        defined as the closing price each week of Qwest Capital Funding        7¼% notes, due February of 2011. The underlying event can also        be defined using an alternative measure, such as the volume        weighted average price during any day. DBAR contingent claims on        government and municipal obligations can be traded in a similar        way.    -   Hybrid Security Prices: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on hybrid securities that        contain both fixed-income and equity features, such as        convertible bond prices. For example, DBAR contingent claims can        be based on an underlying event defined as the closing price        each week of Amazon.com 4¾% convertible bonds due February 2009.        The underlying event can also be defined using an alternative        measure, such as the volume weighted average price during any        day.    -   Interest Rates: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on interest rate measures such        as LIBOR and other money market rates, an index of AAA corporate        bond yields, or any of the fixed income securities listed above.        For example, DBAR contingent claims can be based on an        underlying event defined as the fixing price each week of        3-month LIBOR rates. Alternatively, the underlying event could        be defined as an average of an interest rate over a fixed length        of time, such as a week or month.    -   Foreign Exchange: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on foreign exchange rates. For        example, DBAR contingent claims can be based an underlying event        defined as the exchange rate of the Korean Won on any day.    -   Price & Return Indices: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on a broad variety of financial        instrument price indices, including those for equities (e.g.,        S&P 500), interest rates, commodities, etc. For example, DBAR        contingent claims can be based on an underlying event defined as        the closing price each quarter of the S&P Technology index. The        underlying event can also be defined using an alternative        measure, such as the volume weighted average price during any        day. Alternatively, other index measurements can be used such as        return instead of price.    -   Swaps: Demand-based markets or auctions can be structured to        trade DBAR contingent claims, including, for example, digital        options, based on interest rate swaps and other swap based        transactions. In this example, discussed further in an        embodiment described in Section 9, digital options traded in a        demand-based market or auction are based on an underlying event        defined as the 10 year swap rate at which a fixed 10 year yield        is received against paying a floating 3 month LIBOR rate. The        rate may be determined using a common fixing convention.

Other derivatives on any security or other financial product orinstrument may be used as the underlying instrument for an event ofeconomic significance in a demand-based market or auction. For example,such derivatives can include futures, forwards, swaps, floating ratenotes and other structured financial products. Alternatively, securities(as well as other financial products or instruments) and derivativesthereof can be converted into equivalent DBAR contingent claims (forexample, as in the embodiment discussed in Section 10) and traded as ademand-enabled product alongside DBAR contingent claims in the samedemand-based market or auction.

3.2 DBAR Portfolios

It may be desirable to combine a number of groups of DBAR contingentclaims based on different events into a single portfolio. In this way,traders can invest amounts within the distribution of defined statescorresponding to a single event as well as across the distributions ofstates corresponding to all the groups of contingent claims in theportfolio. In preferred embodiments, the payouts to the amounts investedin this fashion can therefore be a function of a relative comparison ofall the outcome states in the respective groups of DBAR contingentclaims to each other. Such a comparison may be based upon the amountinvested in each outcome state in the distribution for each group ofcontingent claims as well as other qualities, parameters orcharacteristics of the outcome state (e.g., the magnitude of change foreach security underlying the respective groups of contingent claims). Inthis way, more complex and varied payout and return profiles can beachieved using the systems and methods of the present invention. Since apreferred embodiment of a demand reallocation function (DRF) can operateon a portfolio of DBAR contingent claims, such a portfolio is referredto as a DBAR Portfolio, or DBARP. A DBARP is a preferred embodiment ofDBAR contingent claims according to the present invention based on amulti-state, multi-event DRF.

In a preferred embodiment of a DBARP involving different events relatingto different financial products, a DRF is employed in which returns foreach contingent claim in the portfolio are determined by (i) the actualmagnitude of change for each underlying financial product and (ii) howmuch has been invested in each state in the distribution. A large amountinvested in a financial product, such as a common stock, on the longside will depress the returns to defined states on the long side of acorresponding group of DBAR contingent claims. Given the inverserelationship in preferred embodiments between amounts invested in andreturns from a particular state, one advantage to a DBAR portfolio isthat it is not prone to speculative bubbles. More specifically, inpreferred embodiments a massive influx of long side trading, forexample, will increase the returns to short side states, therebyincreasing returns and attracting investment in those states.

The following notation is used to explain further preferred embodimentsof DBARP:

-   -   μ_(i) is the actual magnitude of change for financial product i    -   W_(i) is the amount of successful investments in financial        product i    -   L_(i) is the amount of unsuccessful investments in financial        product i    -   f is the system transaction fee    -   L is the aggregate losses=

$\sum\limits_{i}L_{i}$

-   -   γ_(i) is the normalized returns for successful trades=

$\frac{\mu_{i}}{\sum\limits_{i}{\mu_{i}}}$

-   -   π^(p) _(i) is the payout per value unit invested in financial        product i for a successful investment    -   r^(p) _(i) is the return per unit invested in financial product        i for a successful investment

The payout principle of a preferred embodiment of a DBARP is to returnto a successful investment a portion of aggregate losses scaled by thenormalized return for the successful investment, and to return nothingto unsuccessful investments. Thus, in a preferred embodiment a largeactual return on a relatively lightly traded financial product willbenefit from being allocated a high proportion of the unsuccessfulinvestments.

$\pi_{i}^{p} = \frac{\gamma_{i}*L}{W_{i}}$$r_{L}^{p}:={\frac{\gamma_{i}*L}{W_{i}} - 1}$

As explained below, the correlations of returns across securities isimportant in preferred embodiments to determine payouts and returns in aDBARP.

An example illustrates the operation of a DBARP according to the presentinvention. For purposes of this example, it is assumed that a portfoliocontains two stocks, IBM and MSFT (Microsoft) and that the followinginformation applies (e.g., predetermined termination criteria):

Trading start date: Sep. 1, 1999

Expiration date: Oct. 1, 1999

Current trading period start date: Sep. 1, 1999

Current trading period end date: Sep. 5, 1999

Current date: Sep. 2, 1999

IBM start price: 129

MSFT start price: 96

Both IBM and MSFT Ex-dividends

No transaction fee

In this example, states can be defined so that traders can invest forIBM and MSFT to either depreciate or appreciate over the period. It isalso assumed that the distribution of amounts invested in the variousstates is the following at the close of trading for the current tradingperiod:

Financial Product Depreciate State Appreciate State MSFT $100 million$120 million IBM  $80 million  $65 millionThe amounts invested express greater probability assessments that MSFTwill likely appreciate over the period and IBM will likely depreciate.

For purposes of this example, it is further assumed that on theexpiration date of Oct. 1, 1999, the following actual outcomes forprices are observed:

MSFT: 106 (appreciated by 10.42%)

IBM 127 (depreciated by 1.55%)

In this example, there is $100+$65=$165 million to distribute from theunsuccessful investments to the successful investments, and, for thesuccessful investments, the relative performance of MSFT(10/42/(10.42+1.55)=0.871) is higher than for IBM(1.55/10.42+1.55)=0.229). In a preferred embodiment, 87.1% of theavailable returns is allocated to the successful MSFT traders, with theremainder due the successful IBM traders, and with the following returnscomputed for each state:

-   -   MSFT: $120 million of successful investment produces a payout of        0.871*$165 million=$143.72 million for a return to the        successful traders of

${\frac{{120M} + {143.72M}}{120M} - 1} = {119.77\%}$

-   -   IBM: $80 million in successful investment produces a payout of        (1-0.871)*$165 million=$21.285 million, for a return to the        successful traders of

${\frac{{80M} + {21.285M}}{80M} - 1} = {26.6\%}$

The returns in this example and in preferred embodiments are a functionnot only of the amounts invested in each group of DBAR contingentclaims, but also the relative magnitude of the changes in prices for theunderlying financial products or in the values of the underlying eventsof economic performance. In this specific example, the MSFT tradersreceive higher returns since MSFT significantly outperformed IBM. Inother words, the MSFT longs were “more correct” than the IBM shorts.

The operation of a DBARP is further illustrated by assuming instead thatthe prices of both MSFT and IBM changed by the same magnitude, e.g.,MSFT went up 10%, and IBM went down 10%, but otherwise maintaining theassumptions for this example. In this scenario, $165 million of returnswould remain to distribute from the unsuccessful investments but theseare allocated equally to MSFT and IBM successful investments, or $82.5million to each. Under this scenario the returns are:

MSFT:

${\frac{{120M} + {82.5M}}{120M} - 1} = {68.75\%}$

IBM:

${\frac{{80M} + {82.5M}}{80M} - 1} = {103.125\%}$

The IBM returns in this scenario are 1.5 times the returns to the MFSTinvestments, since less was invested in the IBM group of DBAR contingentclaims than in the MSFT group.

This result confirms that preferred embodiments of the systems andmethods of the present invention provide incentives for traders to makelarge investments, i.e. promote liquidity, where it is needed in orderto have an aggregate amount invested sufficient to provide a fairindication of trader expectations.

The payouts in this example depend upon both the magnitude of change inthe underlying stocks as well as the correlations between such changes.A statistical estimate of these expected changes and correlations can bemade in order to compute expected returns and payouts during trading andat the close of each trading period. While making such an investment maybe somewhat more complicated that in a DBAR range derivative, asdiscussed above, it is still readily apparent to one of skill in the artfrom this specification or from practice of the invention.

The preceding example of a DBARP has been illustrated with eventscorresponding to closing prices of underlying securities. DBARPs of thepresent invention are not so limited and may be applied to any events ofeconomic significance, e.g., interest rates, economic statistics,commercial real estate rentals, etc. In addition, other types of DRFsfor use with DBARPs are apparent to one of ordinary skill in the art,based on this specification or practice of the present invention.

4. Risk Calculations

Another advantage of the groups of DBAR contingent claims according tothe present invention is the ability to provide transparent riskcalculations to traders, market risk managers, and other interestedparties. Such risks can include market risk and credit risk, which arediscussed below.

4.1 Market Risk

Market risk calculations are typically performed so that traders haveinformation regarding the probability distribution of profits and lossesapplicable to their portfolio of active trades. For all tradesassociated with a group of DBAR contingent claims, a trader might wantto know, for example, the dollar loss associated with the bottom fifthpercentile of profit and loss. The bottom fifth percentile correspondsto a loss amount which the trader knows, with a 95% statisticalconfidence, would not be exceeded. For the purposes of thisspecification, the loss amount associated with a given statisticalconfidence (e.g., 95%, 99%) for an individual investment is denoted thecapital-at-risk (“CAR”). In preferred embodiments of the presentinvention, a CAR can be computed not only for an individual investmentbut also for a plurality of investments related to for the same event orfor multiple events.

In the financial industry, there are three common methods that arecurrently employed to compute CAR: (1) Value-at-Risk (“VAR”); (2) MonteCarlo Simulation (“MCS”); and (3) Historical Simulation (“HS”).

4.1.1 Capital-At-Risk Determinations Using Value-At-Risk Techniques

VAR is a method that commonly relies upon calculations of the standarddeviations and correlations of price changes for a group of trades.These standard deviations and correlations are typically computed fromhistorical data. The standard deviation data are typically used tocompute the CAR for each trade individually.

To illustrate the use of VAR with a group of DBAR contingent claims ofthe present invention, the following assumptions are made: (i) a traderhas made a traditional purchase of a stock, say $100 of IBM; (ii) usingpreviously computed standard deviation data, it is determined that theannual standard deviation for IBM is 30%; (iii) as is commonly the case,the price changes for IBM have a normal distribution; and (iv) thepercentile of loss to be used is the bottom fifth percentile. Fromstandard normal tables, the bottom fifth percentile of loss correspondsto approximately 1.645 standard deviations, so the CAR in thisexample—that is, loss for the IBM position that would not be exceededwith 95% statistical confidence—is 30%*1.645*$100, or $49.35. A similarcalculation, using similar assumptions, has been made for a $200position in GM, and the CAR computed for GM is $65.50. If, in thisexample, the computed correlation, ç, between the prices of IBM and GMstock is 0.5, the CAR for the portfolio containing both the IBM and GMpositions may be expressed as:

$\begin{matrix}{{C\; A\; R} = \sqrt{\begin{matrix}{\left( {1.645\alpha_{IBM}\sigma_{IBM}} \right)^{2} + \left( {1.645\alpha_{GM}\sigma_{GM}} \right)^{2} +} \\{2{ϛ1}{.645}\alpha_{IBM}\sigma_{IBM}*1.645\alpha_{GM}\sigma_{GM}}\end{matrix}}} \\{= \sqrt{49.35^{2} + 65.50^{2} + {2*{.5}*49.35*65.5}}} \\{= 99.79}\end{matrix}$

-   -   where α is the investment in dollars, σ is the standard        deviation, and ç is the correlation.        -   These computations are commonly represented in matrix form            as:    -   C is the correlation matrix of the underlying events,    -   w is the vector containing the CAR for each active position in        the portfolio, and    -   w^(T) is the transpose of W.        In preferred embodiments, C is a y×y matrix, where y is the        number of active positions in the portfolio, and where the        elements of C are:    -   c_(i,j)=1 when i=j i.e., has 1's on the diagonal, and otherwise    -   c_(i,j)=the correlation between the ith and jth events

$\begin{matrix}{{C\; A\; R} = \sqrt{w^{T}*C*w}} \\{= \sqrt{\begin{pmatrix}49.35 & 65.5\end{pmatrix}\begin{pmatrix}1 & {.5} \\{.5} & 1\end{pmatrix}\begin{pmatrix}49.35 \\65.5\end{pmatrix}}}\end{matrix}$

In preferred embodiments, several steps implement the VAR methodologyfor a group of DBAR contingent claims of the present invention. Thesteps are first listed, and details of each step are then provided. Thesteps are as follows:

(1) beginning with a distribution of defined states for a group of DBARcontingent claims, computing the standard deviation of returns in valueunits (e.g., dollars) for each investment in a given state;

(2) performing a matrix calculation using the standard deviation ofreturns for each state and the correlation matrix of returns for thestates within the same distribution of states, to obtain the standarddeviation of returns for all investments in a group of DBAR contingentclaims;

(3) adjusting the number resulting from the computation in step (2) foreach investment so that it corresponds to the desired percentile ofloss;

(4) arranging the numbers resulting from step (3) for each distinct DBARcontingent claim in the portfolio into a vector, w, having dimensionequal to the number of distinct DBAR contingent claims;

(5) creating a correlation matrix including the correlation of each pairof the underlying events for each respective DBAR contingent claim inthe portfolio; and

(6) calculating the square root of the product of w, the correlationmatrix created in step (5), and the transpose of w.

The result is CAR using the desired percentile of loss, for all thegroups of DBAR contingent claims in the portfolio.

In preferred embodiments, the VAR methodology of steps (1)-(6) above canbe applied to an arbitrary group of DBAR contingent claims as follows.For purposes of illustrating this methodology, it is assumed that allinvestments are made in DBAR range derivatives using a canonical DRF aspreviously described. Similar analyses apply to other forms of DRFs.

In step (1), the standard deviation of returns per unit of amountinvested for each state i for each group of DBAR contingent claim iscomputed as follows:

$\sigma_{i} = {\sqrt{\frac{T}{T_{i} - 1}} = {\sqrt{\frac{\left( {1 - q_{i}} \right)}{q_{i}}} = \sqrt{r_{i}}}}$

-   -   where σ_(i) is the standard deviation of returns per unit of        amount invested in each state I, T_(i) is the total amount        invested in state i; T is the sum of all amounts invested across        the distribution of states; q_(i) is the implied probability of        the occurrence of state i derived from T and T_(i); and r_(i) is        the return per unit of investment in state i. In this preferred        embodiment, this standard deviation is a function of the amount        invested in each state and total amount invested across the        distributiOn of states, and is also equal to the square root of        the unit return for the state. If α_(i) is the amount invested        in state I, α_(i)*σ_(i) is the standard deviation in units of        the amount invested (e.g., dollars) for each state i.

Step (2) computes the standard deviation for all investments in a groupof DBAR contingent claims. This step (2) begins by calculating thecorrelation between each pair of states for every possible pair withinthe same distribution of states for a group of DBAR contingent claims.For a canonical DRF, these correlations may be computed as follows:

$\begin{matrix}{\rho_{i,j} = {- \frac{\sqrt{T_{i}*T_{j}}}{\sqrt{\left( {T - T_{i}} \right)*\left( {T - T_{j}} \right)}}}} \\{= {- \sqrt{\frac{q_{i}*q_{j}}{\left( {1 - q_{i}} \right)*\left( {1 - q_{j}} \right)}}}} \\{= \frac{- 1}{\sqrt{r_{i}*r_{j}}}} \\{= \frac{- 1}{\sigma_{i}*\sigma_{j}}}\end{matrix}$

-   -   where ρ_(i,j) is the correlation between state i and state j. In        preferred embodiments, the returns to each state are negatively        correlated since the occurrence of one state (a successful        investment) precludes the occurrence of other states        (unsuccessful investments). If there are only two states in the        distribution of states, then T_(j)=T-T_(i) and the correlation        ρ_(i,j) is −1, i.e., an investment in state i is successful and        in state j is not, or vice versa, if i and j are the only two        states. In preferred embodiments where there are more than two        states, the correlation falls in the range between 0 and −1 (the        correlation is exactly 0 if and only if one of the states has        implied probability equal to one). In step (2) of the VAR        methodology, the correlation coefficients ρ_(i,j) are put into a        matrix C_(s) (the subscript s indicating correlation among        states for the same event) which contains a number of rows and        columns equal to the number of defined states for the group of        DBAR contingent claims. The correlation matrix contains 1's        along the diagonal, is symmetric, and the element at the i-th        row and j-th column of the matrix is equal to ρ_(i,j). From        step (1) above, a n×1 vector U is constructed having a dimension        equal to the number of states n, in the group of DBAR contingent        claims, with each element of U being equal to α_(i)*σ_(i). The        standard deviation, w_(k), of returns for all investments in        states within the distribution of states defining the kth group        of DBAR contingent claims can be calculated as follows:

w _(k)=√{square root over (U ^(T) *C _(s) *U)}

Step (3) involves adjusting the previously computed standard deviation,w_(k), for every group of DBAR contingent claims in a portfolio by anamount corresponding to a desired or acceptable percentile of loss. Forpurposes of illustration, it is assumed that investment returns have anormal distribution function; that a 95% statistical confidence forlosses is desirable; and that the standard deviations of returns foreach group of DBAR contingent claims, w_(k), can be multiplied by 1.645,i.e., the number of standard deviations in the standard normaldistribution corresponding to the bottom fifth percentile. A normaldistribution is used for illustrative purposes, and other types ofdistributions (e.g., the Student T distribution) can be used to computethe number of standard deviations corresponding to the any percentile ofinterest. As discussed above, the maximum amount that can be lost inpreferred embodiments of canonical DRF implementation of a group of DBARcontingent claims is the amount invested.

Accordingly, for this illustration the standard deviations w_(k) areadjusted to reflect the constraint that the most that can be lost is thesmaller of (a) the total amount invested and (b) the percentile loss ofinterest associated with the CAR calculation for the group of DBARcontingent claims, i.e.:

$w_{k} = {\min\left( {{1.645*w_{k}},{\sum\limits_{i = {1\mspace{14mu} \ldots \mspace{14mu} n}}\alpha_{i}}} \right)}$

In effect, this updates the standard deviation for each event bysubstituting for it a CAR value that reflects a multiple of the standarddeviation corresponding to an extreme loss percentile (e.g., bottomfifth) or the total invested amount, whichever is smaller.

Step (4) involves taking the adjusted w_(k), as developed in step (4)for each of m groups of DBAR contingent claims, and arranging them intoan y×1 dimensional column vector, w, each element of which containsw_(k), k=1 . . . y.

Step (5) involves the development of a symmetric correlation matrix,C_(e), which has a number of rows and columns equal to the number ofgroups of DBAR contingent claims, y, in which the trader has one or moreinvestments. Correlation matrix C_(e) can be estimated from historicaldata or may be available more directly, such as the correlation matrixamong foreign exchange rates, interest rates, equity indices,commodities, and other financial products available from JP Morgan'sRiskMetrics database. Other sources of the correlation information formatrix C_(e) are known to those of skill in the art. Along the diagonalsof the correlation matrix C_(e) are 1's and the entry at the i-th rowand j-th column of the matrix contains the correlation between the i-thand j-th events which define the i-th and j-th DBAR contingent claim forall such possible pairs among the m active groups of DBAR contingentclaims in the portfolio.

In Step (6), the CAR for the entire portfolio of m groups of DBARcontingent claims is found by performing the following matrixcomputation, using each w_(k) from step (4) arrayed into vector w andits transpose w^(T):

CAR=√{square root over (w ^(T) *C _(e) *w)}

This CAR value for the portfolio of groups of DBAR contingent claims isan amount of loss that will not be exceeded with the associatedstatistical confidence used in Steps (1)-(6) above (e.g., in thisillustration, 95%).

Example 4.1.1-1 VAR-Based CAR Calculation

An example further illustrates the calculation of a VAR-based CAR for aportfolio containing two groups of DBAR range derivative contingentclaims (i.e., y=2) with a canonical DRF on two common stocks, IBM andGM. For this example, the following assumptions are made: (i) for eachof the two groups of DBAR contingent claims, the relevant underlyingevent upon which the states are defined is the respective closing priceof each stock one month forward; (ii) there are only three statesdefined for each event: “low”, “medium”, and “high,” corresponding toranges of possible closing prices on that date; (iii) the posted returnsfor IBM and GM respectively for the three respective states are, in U.S.dollars, (4, 0.6667, 4) and (2.333, 1.5, 2.333); (iv) the exchange feeis zero; (v) for the IBM group of contingent claims, the trader has onedollar invested in the state “low”, three dollars invested in the state“medium,” and two dollars invested in the state “high”; (vi) for the GMgroup of contingent claims, the trader has a single investment in theamount of one dollar in the state “medium”; (vii) the desired oracceptable percentile of loss in the fifth percentile, assuming a normaldistribution; and (viii) the estimated correlation of the price changesof IBM and GM is 0.5 across the distribution of states for each stock.

Steps (1)-(6), described above, are used to implement VAR in order tocompute CAR for this example. From Step (1), the standard deviations ofstate returns per unit of amount invested in each state for the IBM andGM groups of contingent claims are, respectively, (2, 0.8165, 2) and(1.5274, 1.225, 1.5274). In further accordance with Step (1) above, theamount invested in each state in the respective group of contingentclaims, α_(i); is multiplied by the previously calculated standarddeviation of state returns per investment, σ_(i), so that the standarddeviation of returns per state in dollars for each claim equals, for theIBM group: (2, 2.4495, 4) and, for the GM group, (0,1.225, 0).

In accordance with Step (2) above, for each of the two groups of DBARcontingent claims in this example, a correlation matrix between any pairof states, C_(s), is constructed, as follows:

$C_{s}^{IBM} = \begin{matrix}1 & {- {.6124}} & {- {.25}} \\{.6124} & 1 & {- {.6124}} \\{- {.25}} & {- {.6124}} & 1\end{matrix}$ $C_{s}^{GM} = \begin{matrix}1 & {- {.5345}} & {- {.4286}} \\{- {.5345}} & 1 & {- {.5345}} \\{- {.4286}} & {- {.5345}} & 1\end{matrix}$

where the left matrix is the correlation between each pair of statereturns for the IBM group of contingent claims and the right matrix isthe corresponding matrix for the GM group of contingent claims.

Also according to step (2) above, for each of the two groups ofcontingent claims, the standard deviation of returns per state indollars, α_(i)σ_(i), for each investment in this example can be arrangedin a vector with dimension equal to three (i.e., the number of states):

$U_{IBM} = \begin{matrix}2 \\2.4495 \\4\end{matrix}$ $U_{GM} = \begin{matrix}0 \\1.225 \\0\end{matrix}$

where the vector on the left contains the standard deviation in dollarsof returns per state for the IBM group of contingent claims, and thevector on the right contains the corresponding information for the GMgroup of contingent claims. Further in accordance with Step (2) above, amatrix calculation can be performed to compute the total standarddeviation for all investments in each of the two groups of contingentclaims, respectively:

w ₁=√{square root over (U _(iBM) ^(T) *C _(s) ^(IBM) *U _(iBM))}=2

w ₂=√{square root over (U _(GM) ^(T) *C _(s) ^(GM) *U _(GM))}=1.225

where the quantity on the left is the standard deviation for allinvestments in the distribution of the IBM group of contingent claims,and the quantity on the right is the corresponding standard deviationfor the GM group of contingent claims.

In accordance with step (3) above, w₁ and w₂ are adjusted by multiplyingeach by 1.645 (corresponding to a CAR loss percentile of the bottomfifth percentile assuming a normal distribution) and then taking thelower of (a) that resulting value and (b) the maximum amount that can belost, i.e., the amount invested in all states for each group ofcontingent claims:

w ₁=min(2*1.645,6)=3.29 w ₂=min(2*1.225,1)=1

where the left quantity is the adjusted standard deviation of returnsfor all investments across the distribution of the IBM group ofcontingent claims, and the right quantity is the corresponding amountinvested in the GM group of contingent claims. These two quantities, w₁and w₂, are the CAR values for the individual groups of DBAR contingentclaims respectively, corresponding to a statistical confidence of 95%.In other words, if the normal distribution assumptions that have beenmade with respect to the state returns are valid, then a trader, forexample, could be 95% confident that losses on the IBM groups ofcontingent claims would not exceed $3.29.

Proceeding now with Step (4) in the VAR process described above, thequantities w₁ and w₂ are placed into a vector which has a dimension oftwo, equal to the number of groups of DBAR contingent claims in theillustrative trader's portfolio:

$w = \begin{matrix}3.29 \\1\end{matrix}$

According to Step (5), a correlation matrix C_(e) with two rows and twocolumns, is either estimated from historical data or obtained from someother source (e.g., RiskMetrics), as known to one of skill in the art.Consistent with the assumption for this illustration that the estimatedcorrelation between the price changes of IBM and GM is 0.5, thecorrelation matrix for the underlying events is as follows:

$C_{e} = \begin{matrix}1 & {.5} \\{.5} & 1\end{matrix}$

Proceeding with Step (6), a matrix multiplication is performed by pre-and post-multiplying C_(e) by the transpose of w and by w, and takingthe square root of the resulting product:

CAR=√{square root over (w ^(T) *C _(e) *w)}=3.8877

This means that for the portfolio in this example, comprising the threeinvestments in the IBM group of contingent claims and the singleinvestment in the GM group of contingent claims, the trader can have a95% statistical confidence he will not have losses in excess of $3.89.

4.1.2 Capital-At-Risk Determinations Using Monte Carlo SimulationTechniques

Monte Carlo Simulation (“MCS”) is another methodology that is frequentlyused in the financial industry to compute CAR. MCS is frequently used tosimulate many representative scenarios for a given group of financialproducts, compute profits and losses for each representative scenario,and then analyze the resulting distribution of scenario profits andlosses. For example, the bottom fifth percentile of the distribution ofthe scenario profits and losses would correspond to a loss for which atrader could have a 95% confidence that it would not be exceeded. In apreferred embodiment, the MCS methodology can be adapted for thecomputation of CAR for a portfolio of DBAR contingent claims as follows.

Step (1) of the MCS methodology involves estimating the statisticaldistribution for the events underlying the DBAR contingent claims usingconventional econometric techniques, such as GARCH. If the portfoliobeing analyzed has more than one group of DBAR contingent claim, thenthe distribution estimated will be what is commonly known as amultivariate statistical distribution which describes the statisticalrelationship between and among the events in the portfolio. For example,if the events are underlying closing prices for stocks and stock pricechanges have a normal distribution, then the estimated statisticaldistribution would be a multivariate normal distribution containingparameters relevant for the expected price change for each stock, itsstandard deviation, and correlations between every pair of stocks in theportfolio. Multivariate statistical distribution is typically estimatedfrom historical time series data on the underlying events (e.g., historyof prices for stocks) using conventional econometric techniques.

Step (2) of the MCS methodology involves using the estimated statisticaldistribution of Step (1) in order to simulate the representativescenarios. Such simulations can be performed using simulation methodscontained in such reference works as Numerical Recipes in C or by usingsimulation software such as @Risk package available from Palisade, orusing other methods known to one of skill in the art. For each simulatedscenario, the DRF of each group of DBAR contingent claims in theportfolio determines the payouts and profits and losses on the portfoliocomputed.

Using the above two stock example involving GM and IBM used above todemonstrate VAR techniques for calculating CAR, a scenario simulated byMCS techniques might be “High” for IBM and “Low” for GM, in which casethe trader with the above positions would have a four dollar profit forthe IBM contingent claim and a one dollar loss for the GM contingentclaim, and a total profit of three dollars. In step (2), many suchscenarios are generated so that a resulting distribution of profit andloss is obtained. The resulting profits and losses can be arranged intoascending order so that, for example, percentiles corresponding to anygiven profit and loss number can be computed. A bottom fifth percentile,for example, would correspond to a loss for which the trader could be95% confident would not be exceeded, provided that enough scenarios havebeen generated to provide an adequate representative sample. This numbercould be used as the CAR value computed using MCS for a group of DBARcontingent claims. Additionally, statistics such as average profit orloss, standard deviation, skewness, kurtosis and other similarquantities can be computed from the generated profit and lossdistribution, as known by one of skill in the art.

4.1.3 Capital-At-Risk Determination Using Historical SimulationTechniques

Historical Simulation (“HS”) is another method used to compute CARvalues. HS is comparable to that of MCS in that it relies upon the useof representative scenarios in order to compute a distribution of profitand loss for a portfolio. Rather than rely upon simulated scenarios froman estimated probability distribution, however, HS uses historical datafor the scenarios. In a preferred embodiment, HS can be adapted to applyto a portfolio of DBAR contingent claims as follows.

Step (1) involves obtaining, for each of the underlying eventscorresponding to each group of DBAR contingent claims, a historical timeseries of outcomes for the events. For example, if the events are stockclosing prices, time series of closing prices for each stock can beobtained from a historical database such as those available fromBloomberg, Reuters, or Datastream or other data sources known to someoneof skill in the art.

Step (2) involves using each observation in the historical data fromStep (1) to compute payouts using the DRF for each group of DBARcontingent claims in the portfolio. From the payouts for each group foreach historical observation, a portfolio profit and loss can becomputed. This results in a distribution of profits and lossescorresponding to the historical scenarios, i.e., the profit and lossthat would have been obtained had the trader held the portfoliothroughout the period covered by the historical data sample.

Step (3) involves arranging the values for profit and loss from thedistribution of profit and loss computed in Step (2) in ascending order.A profit and loss can therefore be computed corresponding to anypercentile in the distribution so arranged, so that, for example, a CARvalue corresponding to a statistical confidence of 95% can be computedby reference to the bottom fifth percentile.

4.2 Credit Risk

In preferred embodiments of the present invention, a trader may makeinvestments in a group of DBAR contingent claims using a margin loan. Inpreferred embodiments of the present invention implementing DBAR digitaloptions, an investor may make an investment with a profit and lossscenario comparable to a sale of a digital put or call option and thushave some loss if the option expires “in the money,” as discussed inSection 6, below. In preferred embodiments, credit risk may be measuredby estimating the amount of possible loss that other traders in thegroup of contingent claims could suffer owing to the inability of agiven trader to repay a margin loan or otherwise cover a loss exposure.For example, a trader may have invested $1 in a given state for a groupof DBAR contingent claims with $0.50 of margin. Assuming a canonical DRFfor this example, if the state later fails to occur, the DRF collects $1from the trader (ignoring interest) which would require repayment of themargin loan. As the trader may be unable to repay the margin loan at therequired time, the traders with successful trades may potentially not beable to receive the full amounts owing them under the DRF, and maytherefore receive payouts lower than those indicated by the finalizedreturns for a given trading period for the group of contingent claims.Alternatively, the risk of such possible losses due to credit risk maybe insured, with the cost of such insurance either borne by the exchangeor passed on to the traders. One advantage of the system and method ofthe present invention is that, in preferred embodiments, the amount ofcredit risk associated with a group of contingent claims can readily becalculated.

In preferred embodiments, the calculation of credit risk for a portfolioof groups of DBAR contingent claims involves computing acredit-capital-at-risk (“CCAR”) figure in a manner analogous to thecomputation of CAR for market risk, as described above.

The computation of CCAR involves the use of data related to the amountof margin used by each trader for each investment in each state for eachgroup of contingent claims in the portfolio, data related to theprobability of each trader defaulting on the margin loan (which cantypically be obtained from data made available by credit ratingagencies, such as Standard and Poors, and data related to thecorrelation of changes in credit ratings or default probabilities forevery pair of traders (which can be obtained, for example, from JPMorgan's CreditMetrics database).

In preferred embodiments, CCAR computations can be made with varyinglevels of accuracy and reliability. For example, a calculation of CCARthat is substantially accurate but could be improved with more data andcomputational effort may nevertheless be adequate, depending upon thegroup of contingent claims and the desires of traders for credit riskrelated information. The VAR methodology, for example, can be adapted tothe computation of CCAR for a group of DBAR contingent claims, althoughit is also possible to use MCS and HS related techniques for suchcomputations. The steps that can be used in a preferred embodiment tocompute CCAR using VAR-based, MCS-based, and HS-based methods aredescribed below.

4.2.1 CCAR Method for DBAR Contingent Claims Using the VAR-basedMethodology

Step (i) of the VAR-based CCAR methodology involves obtaining, for eachtrader in a group of DBAR contingent claims, the amount of margin usedto make each trade or the amount of potential loss exposure from tradeswith profit and loss scenarios comparable to sales of options inconventional markets.

Step (ii) involves obtaining data related to the probability of defaultfor each trader who has invested in the groups of DBAR contingentclaims. Default probabilities can be obtained from credit ratingagencies, from the JP Morgan CreditMetrics database, or from othersources as known to one of skill in the art. In addition to defaultprobabilities, data related to the amount recoverable upon default canbe obtained. For example, an AA-rated trader with $1 in margin loans maybe able to repay $0.80 dollars in the event of default.

Step (iii) involves scaling the standard deviation of returns in unitsof the invested amounts. This scaling step is described in step (1) ofthe VAR methodology described above for estimating market risk. Thestandard deviation of each return, determined according to Step (1) ofthe VAR methodology previously described, is scaled by (a) thepercentage of margin [or loss exposure] for each investment; (b) theprobability of default for the trader; and (c) the percentage notrecoverable in the event of default.

Step (iv) of this VAR-based CCAR methodology involves taking from step(iii) the scaled values for each state for each investment andperforming the matrix calculation described in Step (2) above for theVAR methodology for estimating market risk, as described above. In otherwords, the standard deviations of returns in units of invested amountswhich have been scaled as described in Step (iii) of this CCARmethodology are weighted according to the correlation between eachpossible pair of states (matrix C_(s), as described above). Theresulting number is a credit-adjusted standard deviation of returns inunits of the invested amounts for each trader for each investment on theportfolio of groups of DBAR contingent claims. For a group of DBARcontingent claims, the standard deviations of returns that have beenscaled in this fashion are arranged into a vector whose dimension equalsthe number of traders.

Step (v) of this VAR-based CCAR methodology involves performing a matrixcomputation, similar to that performed in Step (5) of the VARmethodology for CAR described above. In this computation, the vector ofcredit-scaled standard deviations of returns from step (iv) are used topre- and post-multiply a correlation matrix with rows and columns equalto the number of traders, with 1's along the diagonal, and with theentry at row i and column j containing the statistical correlation ofchanges in credit ratings described above. The square root of theresulting matrix multiplication is an approximation of the standarddeviation of losses, due to default, for all the traders in a group ofDBAR contingent claims. This value can be scaled by a number of standarddeviations corresponding to a statistical confidence of thecredit-related loss not to be exceeded, as discussed above.

In a preferred embodiment, any given trader may be omitted from a CCARcalculation. The result is the CCAR facing the given trader due to thecredit risk posed by other traders who have invested in a group of DBARcontingent claims. This computation can be made for all groups of DBARcontingent claims in which a trader has a position, and the resultingnumber can be weighted by the correlation matrix for the underlyingevents, C_(e), as described in Step (5) for the VAR-based CARcalculation. The result corresponds to the risk of loss posed by thepossible defaults of other traders across all the states of all thegroups of DBAR contingent claims in a trader's portfolio.

4.2.2 CCAR Method for DBAR Contingent Claims Using the Monte CarloSimulation (MCS) Methodology

As described above, MCS methods are typically used to simulaterepresentative scenarios for a given group of financial products,compute profits and losses for each representative scenario, thenanalyze the resulting distribution of scenario profits and losses. Thescenarios are designed to be representative in that they are supposed tobe based, for instance, on statistical distributions which have beenestimated, typically using econometric time series techniques, to have agreat degree of relevance for the future behavior of the financialproducts. A preferred embodiment of MCS methods to estimate CCAR for aportfolio of DBAR contingent claims of the present invention, involvestwo steps, as described below.

Step (i) of the MCS methodology is to estimate a statisticaldistribution of the events of interest. In computing CCAR for a group ofDBAR contingent claims, the events of interest may be both the primaryevents underlying the groups of DBAR contingent claims, including eventsthat may be fitted to multivariate statistical distributions to computeCAR as described above, as well as the events related to the default ofthe other investors in the groups of DBAR contingent claims. Thus, in apreferred embodiment, the multivariate statistical distribution to beestimated relates to the market events (e.g., stock price changes,changes in interest rates) underlying the groups of DBAR contingentclaims being analyzed as well as the event that the investors in thosegroups of DBAR contingent claims, grouped by credit rating orclassification will be unable to repay margin loans for losinginvestments.

For example, a multivariate statistical distribution to be estimatedmight assume that changes in the market events and credit ratings orclassifications are jointly normally distributed. Estimating such adistribution would thus entail estimating, for example, the mean changesin the underlying market events (e.g., expected changes in interestrates until the expiration date), the mean changes in credit ratingsexpected until expiration, the standard deviation for each market eventand credit rating change, and a correlation matrix containing all of thepairwise correlations between every pair of events, including market andcredit event pairs. Thus, a preferred embodiment of MCS methodology asit applies to CCAR estimation for groups of DBAR contingent claims ofthe present invention typically requires some estimation as to thestatistical correlation between market events (e.g., the change in theprice of a stock issue) and credit events (e.g., whether an investorrated A− by Standard and Poors is more likely to default or bedowngraded if the price of a stock issue goes down rather than up).

It is sometimes difficult to estimate the statistical correlationsbetween market-related events such as changes in stock prices andinterest rates, on the one hand, and credit-related events such ascounterparty downgrades and defaults, on the other hand. Thesedifficulties can arise due to the relative infrequency of creditdowngrades and defaults. The infrequency of such credit-related eventsmay mean that statistical estimates used for MCS simulation can only besupported with low statistical confidence. In such cases, assumptionscan be employed regarding the statistical correlations between themarket and credit-related events. For example, it is not uncommon toemploy sensitivity analysis with regard to such correlations, i.e., toassume a given correlation between market and credit-related events andthen vary the assumption over the entire range of correlations from −1to 1 to determine the effect on the overall CCAR.

A preferred approach to estimating correlation between events is to usea source of data with regard to credit-related events that does nottypically suffer from a lack of statistical frequency. Two methods canbe used in this preferred approach. First, data can be obtained thatprovide greater statistical confidence with regard to credit-relatedevents. For example, expected default frequency data can be purchasedfrom such companies as KMV Corporation. These data supply probabilitiesof default for various parties that can be updated as frequently asdaily. Second, more frequently observed default probabilities can beestimated from market interest rates. For example, data providers suchas Bloomberg and Reuters typically provide information on the additionalyield investors require for investments in bonds of varying creditratings, e.g., AAA, AA, A, A−. Other methods are readily available toone skilled in the art to provide estimates regarding defaultprobabilities for various entities. Such estimates can be made asfrequently as daily so that it is possible to have greater statisticalconfidence in the parameters typically needed for MCS, such as thecorrelation between changes in default probabilities and changes instock prices, interest rates, and exchange rates.

The estimation of such correlations is illustrated assuming two groupsof DBAR contingent claims of interest, where one group is based upon theclosing price of IBM stock in three months, and the other group is basedupon the closing yield of the 30-year U.S. Treasury bond in threemonths. In this illustration, it is also assumed that the counterpartieswho have made investments on margin in each of the groups can be dividedinto five distinct credit rating classes. Data on the daily changes inthe price of IBM and the bond yield may be readily obtained from suchsources as Reuters or Bloomberg. Frequently changing data on theexpected default probability of investors can be obtained, for example,from KMV Corporation, or estimated from interest rate data as describedabove. As the default probability ranges between 0 and 1, a statisticaldistribution confined to this interval is chosen for purposes of thisillustration. For example, for purposes of this illustration, it can beassumed that the expected default probability of the investors follows alogistic distribution and that the joint distribution of changes in IBMstock and the 30-year bond yield follows a bivariate normaldistribution. The parameters for the logistic distribution and thebivariate normal distribution can be estimated using econometrictechniques known to one skilled in the art.

Step (ii) of a MCS technique, as it may be applied to estimating CCARfor groups of DBAR contingent claims, involves the use of themultivariate statistical distributions estimated in Step (i) above inorder to simulate the representative scenarios. As described above, suchsimulations can be performed using methods and software readilyavailable and known to those of skill in the art. For each simulatedscenario, the simulated default rate can be multiplied by the amount oflosses an investor faces based upon the simulated market changes and themargin, if any, the investor has used to make losing investments. Theproduct represents an estimated loss rate due to investor defaults. Manysuch scenarios can be generated so that a resulting distribution ofcredit-related expected losses can be obtained. The average value of thedistribution is the mean loss. The lowest value of the top fifthpercentile of the distribution, for example, would correspond to a lossfor which a given trader could be 95% confident would not be exceeded,provided that enough scenarios have been generated to provide astatistically meaningful sample. In preferred embodiments, the selectedvalue in the distribution, corresponding to a desired or adequateconfidence level, is used as the CCAR for the groups of DBAR contingentclaims being analyzed.

4.2.3 CCAR Method for DBAR Contingent Claims Using the HistoricalSimulation (“HS”) Methodology

As described above, Historical Simulation (HS) is comparable to MCS forestimating CCAR in that HS relies on representative scenarios in orderto compute a distribution of profit and loss for a portfolio of groupsof DBAR contingent claim investments. Rather than relying on simulatedscenarios from an estimated multivariate statistical distribution,however, HS uses historical data for the scenarios. In a preferredembodiment, HS methodology for calculating CCAR for groups of DBARcontingent claims uses three steps, described below.

Step (i) involves obtaining the same data for the market-related eventsas described above in the context of CAR. In addition, to use HS toestimate CCAR, historical time series data are also used forcredit-related events such as downgrades and defaults. As such data aretypically rare, methods described above can be used to obtain morefrequently observed data related to credit events. For example, in apreferred embodiment, frequently-observed data on expected defaultprobabilities can be obtained from KMV Corporation. Other means forobtaining such data are known to those of skill in the art.

Step (ii) involves using each observation in the historical data fromthe previous step (i) to compute payouts using the DRF for each group ofDBAR contingent claims being analyzed. The amount of margin to be repaidfor the losing trades, or the loss exposure for investments with profitand loss scenarios comparable to digital option “sales,” can then bemultiplied by the expected default probability to use HS to estimateCCAR, so that an expected loss number can be obtained for each investorfor each group of contingent claims. These losses can be summed acrossthe investment by each trader so that, for each historical observationdata point, an expected loss amount due to default can be attributed toeach trader. The loss amounts can also be summed across all theinvestors so that a total expected loss amount can be obtained for allof the investors for each historical data point.

Step (iii) involves arranging, in ascending order, the values of lossamounts summed across the investors for each data point from theprevious step (ii). An expected loss amount due to credit-related eventscan therefore be computed corresponding to any percentile in thedistribution so arranged. For example, a CCAR value corresponding to a95% statistical confidence level can be computed by reference to 95^(th)percentile of the loss distribution.

5. Liquidity and Price/Quantity Relationships

In the trading of contingent claims, whether in traditional markets orusing groups of DBAR contingent claims of the present invention, it isfrequently useful to distinguish between the fundamental value of theclaim, on the one hand, as determined by market expectations,information, risk aversion and financial holdings of traders, and thedeviations from such value due to liquidity variations, on the otherhand. For example, the fair fundamental value in the traditional swapmarket for a five-year UK swap (i.e., swapping fixed interest forfloating rate payments based on UK LIBOR rates) might be 6.79% with a 2basis point bid/offer (i.e., 6.77% receive, 6.81% pay). A large traderwho takes the market's fundamental mid-market valuation of 6.79% ascorrect or fair might want to trade a swap for a large amount, such as750 million pounds. In light of likely liquidity available according tocurrent standards of the traditional market, the large amount of thetransaction could reduce the likely offered rate to 6.70%, which is afull 7 basis points lower than the average offer (which is probablyapplicable to offers of no more than 100 million pounds) and 9 basispoints away from the fair mid-market value.

The difference in value between a trader's position at the fair ormid-market value and the value at which the trade can actually becompleted, i.e. either the bid or offer, is usually called the liquiditycharge. For the illustrative five-year UK swap, a 1 basis pointliquidity charge is approximately equal to 0.04% of the amount traded,so that a liquidity charge of 9 basis points equals approximately 2.7million pounds. If no new information or other fundamental shocksintrude into or “hit” the market, this liquidity charge to the trader isalmost always a permanent transaction charge for liquidity—one that alsomust be borne when the trader decides to liquidate the large position.Additionally, there is no currently reliable way to predict, in thetraditional markets, how the relationship between price and quantity maydeviate from the posted bid and offers, which are usually applicableonly to limited or representative amounts. Price and quantityrelationships can be highly variable, therefore, due to liquidityvariations. Those relationships can also be non-linear. For instance, itmay cost more than twice as much, in terms of a bid/offer spread, totrade a second position that is only twice as large as a first position.

From the point of view of liquidity and transactions costs, groups ofDBAR contingent claims of the present invention offer advantagescompared to traditional markets. In preferred embodiments, therelationship between price (or returns) and quantity invested (i.e.,demanded) is determined mathematically by a DRF. In a preferredembodiment using a canonical DRF, the implied probability q_(i) for eachstate i increases, at a decreasing rate, with the amount invested inthat state:

$q_{i} = \frac{T_{i}}{T}$$\frac{\partial q_{i}}{\partial T_{i}} = \frac{T - T_{i}}{T^{2}}$$\frac{\partial^{2}q_{i}}{\partial T_{i}^{2}} = {{- 2}*\frac{T - T_{i}}{T^{3}}}$$\frac{\partial q_{i}}{\partial T_{j,{j \neq i}}} = {{- \frac{T_{i}}{T^{2}}} = {- \frac{q_{i}}{T}}}$

where T is the total amount invested across all the states of the groupof DBAR contingent claims and T_(i) is the amount invested in the statei. As a given the amount gets very large, the implied probability ofthat state asymptotically approaches one. The last expressionimmediately above shows that there is a transparent relationship,available to all traders, between implied probabilities and the amountinvested in states other than a given state i. The expression shows thatthis relationship is negative, i.e., as amounts invested in other statesincrease, the implied probability for the given state i decreases.Since, in preferred embodiments of the present invention, addinginvestments to states other than the given state is equivalent toselling the given state in the market, the expression for

$\frac{\partial q_{i}}{\partial T_{j,{j \neq i}}}$

above shows how, in a preferred embodiment, the implied probability forthe given state changes as a quantity for that state is up for sale,i.e., what the market's “bid” is for the quantity up for sale. Theexpression for

$\frac{\partial q_{i}}{\partial T_{i}}$

above shows, in a preferred embodiment, how the probability for thegiven state changes when a given quantity is demanded or desired to bepurchased, i.e., what the market's “offer” price is to purchasers of thedesired quantity.

In a preferred embodiment, for each set of quantities invested in thedefined states of a group of DBAR contingent claims, a set of bid andoffer curves is available as a function of the amount invested.

In the groups of DBAR contingent claims of the present invention, thereare no bids or offers per se. The mathematical relationships above areprovided to illustrate how the systems and methods of the presentinvention can, in the absence of actual bid/offer relationships, providegroups of DBAR contingent claims with some of the functionality ofbid/offer relationships.

Economists usually prefer to deal with demand and cross-demandelasticities, which are the percentage changes in prices due topercentage changes in quantity demanded for a given good (demandelasticity) or its substitute (cross-demand elasticity). In preferredembodiments of the systems and methods of the present invention, andusing the notation developed above,

${\frac{\Delta \; q_{i}}{q_{i}}/\frac{\Delta \; T_{i}}{T_{i}}} = {1 - q_{i}}$${\frac{\Delta \; q_{i}}{q_{i}}/\frac{\Delta \; T_{j}}{T_{j}}} = {- q_{j}}$

The first of the expressions immediately above shows that smallpercentage changes in the amount invested in state i have a decreasingpercentage effect on the implied probability for state I, as state ibecomes more likely (i.e., as q_(i) increases to 1). The secondexpression immediately above shows that a percentage change in theamount invested in a state j other than state i will decrease theimplied probability for state i in proportion to the implied probabilityfor the other state j.

In preferred embodiments, in order to effectively “sell” a state,traders need to invest or “buy” complement states, i.e., states otherthan the one they wish to “sell.” Thus, in a preferred embodimentinvolving a group of DBAR claims with two states, a “seller” of state 1will “buy” state 2, and vice versa. In order to “sell” state 1, state 2needs to be “bought” in proportion to the ratio of the amount investedin state 2 to the amount invested in state 1. In a state distributionwhich has more than two states, the “complement” for a given state to be“sold” are all of the other states for the group of DBAR contingentclaims. Thus, “selling” one state involves “buying” a multi-stateinvestment, as described above, for the complement states.

Viewed from this perspective, an implied offer is the resulting effecton implied probabilities from making a small investment in a particularstate. Also from this perspective, an implied bid is the effect onimplied probabilities from making a small multi-state investment incomplement states. For a given state in a preferred embodiment of agroup of DBAR contingent claims, the effect of an invested amount onimplied probabilities can be stated as follows:

${{Implied}\mspace{14mu} {``{Bid}"}} = {q_{i} - {\frac{\left( {1 - q_{i}} \right)}{T}*\Delta \; T_{i}}}$${{Implied}\mspace{14mu} {``{Offer}"}} = {q_{i} + {q_{i}*\left( {\frac{1}{T_{i}} - \frac{1}{T}} \right)*\Delta \; T_{i}}}$

where ΔT_(i) (considered here to be small enough for a first-orderapproximation) is the amount invested for the “bid” or “offer.” Theseexpressions for implied “bid” and implied “offer” can be used forapproximate computations. The expressions indicate how possibleliquidity effects within a group of DBAR contingent claims can be castin terms familiar in traditional markets. In the traditional markets,however, there is no ready way to compute such quantities for any givenmarket.

The full liquidity effect—or liquidity response function—between twostates in a group of DBAR contingent claims can be expressed asfunctions of the amounts invested in a given state, T_(i), and amountsinvested in the complement states, denoted T^(c) _(i), as follows:

${{Implied}\mspace{14mu} {``{Bid}"}\mspace{14mu} {Demand}\mspace{14mu} {Response}\mspace{14mu} {q_{i}^{B}\left( {\Delta \; T_{i}} \right)}} = \frac{T_{i}}{T_{i} + T_{i}^{c} + {\Delta \; T_{i}*\left( \frac{T_{i}^{c}}{T_{i} - {\Delta \; T_{i}}} \right)}}$$\mspace{79mu} {{{Implied}\mspace{14mu} {``{Offer}"}\mspace{14mu} {Demand}\mspace{14mu} {Response}\mspace{14mu} {q_{i}^{O}\left( {\Delta \; T_{i}} \right)}} = \frac{T_{i} + {\Delta \; T_{i}}}{T_{i} + T_{i}^{c} + {\Delta \; T_{i}}}}$

The implied “bid” demand response function shows the effect on theimplied state probability of an investment made to hedge an investmentof size ΔT_(i). The size of the hedge investment in the complementstates is proportional to the ratio of investments in the complementstates to the amount of investments in the state or states to be hedged,excluding the investment to be hedged (i.e., the third term in thedenominator). The implied “offer” demand response function above showsthe effect on the implied state probability from an incrementalinvestment of size ΔT_(i) in a particular defined state.

In preferred embodiments of systems and methods of the presentinvention, only the finalized returns for a given trading period areapplicable for computing payouts for a group of DBAR contingent claims.Thus, in preferred embodiments, unless the effect of a trade amount onreturns is permanent, i.e., persists through the end of a tradingperiod, a group of DBAR contingent claims imposes no permanent liquiditycharge, as the traditional markets typically do. Accordingly, inpreferred embodiments, traders can readily calculate the effect onreturns from investments in the DBAR contingent claims, and unless thesecalculated effects are permanent, they will not affect closing returnsand can, therefore, be ignored in appropriate circumstances. In otherwords, investing in a preferred embodiment of a group of DBAR contingentclaims does not impose a permanent liquidity charge on traders forexiting and entering the market, as the traditional markets typicallydo.

The effect of a large investment may, of course, move intra-tradingperiod returns in a group of DBAR contingent claims as indicated by theprevious calculations. In preferred embodiments, these effects couldwell be counteracted by subsequent investments that move the market backto fair value (in the absence of any change in the fundamental or fairvalue). In traditional markets, by contrast, there is usually a “tollbooth” effect in the sense that a toll or change is usually exactedevery time a trader enters and exits the market. This toll is largerwhen there is less “traffic” or liquidity and represents a permanentloss to the trader. By contrast, other than an exchange fee, inpreferred embodiments of groups of DBAR contingent claims, there is nosuch permanent liquidity tax or toll for market entry or exit.

Liquidity effects may be permanent from investments in a group of DBARcontingent claims if a trader is attempting to make a relatively verylarge investment near the end of a trading period, such that the marketmay not have sufficient time to adjust back to fair value. Thus, inpreferred embodiments, there should be an inherent incentive not to holdback large investments until the end of the trading period, therebyproviding incentives to make large investments earlier, which isbeneficial overall to liquidity and adjustment of returns. Nonetheless,a trader can readily calculate the effects on returns to a investmentwhich the trader thinks might be permanent (e.g., at the end of thetrading period), due to the effect on the market from a large investmentamount.

For example, in the two period hedging example (Example 3.1.19) above,it was assumed that the illustrated trader's investments had no materialeffect on the posted returns, in other words, that this trader was a“price taker.” The formula for the hedge trade H in the second period ofthat example above reflects this assumption. The following equivalentexpression for H takes account of the possibly permanent effect that alarge trade investment might have on the closing returns (because, forexample, the investment is made very close to the end of the tradingperiod):

$H = \frac{P_{t} - T_{t + 1} + \sqrt{T_{t + 1}^{2} - {2*T_{t + 1}*P_{t}} + P_{t}^{2} + {4*P_{t}*T_{t + 1}^{c}}}}{2}$where P_(t) = α_(t) * (1 + r_(t))

in the notation used in Example 3.1.19, above, and T_(t+1) is the totalamount invested in period t+1 and T^(c) _(t+1) is the amount invested inthe complement state in period t+1. The expression for H is thequadratic solution which generates a desired payout, as described abovebut using the present notation. For example, if $1 billion is the totalamount, T, invested in trading period 2, then, according to the aboveexpressions, the hedge trade investment assuming a permanent effect onreturns is $70.435 million compared to $70.18755 million in Example3.1.19. The amount of profit and loss locked-in due to the new hedge is$1.232 million, compared to $1.48077 in Example 3.1.19. The differencerepresents the liquidity effect, which even in the example where theinvested notional is 10% of the total amount invested, is quitereasonable in a market for groups of DBAR contingent claims. There is noready way to estimate or calculate such liquidity effects in traditionalmarkets.

6. Dbar Digital Options Exchange

In a preferred embodiment, the DBAR methods and systems of the presentinvention may be used to implement financial products known as digitaloptions and to facilitate an exchange in such products. A digital option(sometimes also known as a binary option) is a derivative security whichpays a fixed amount should specified conditions be met (such as theprice of a stock exceeding a given level or “strike” price) at theexpiration date. If the specified conditions are met, a digital optionis often characterized as finishing “in the money.” A digital calloption, for example, would pay a fixed amount of currency, say onedollar, should the value of the underlying security, index, or variableupon which the option is based expire at or above the strike price ofthe call option. Similarly, a digital put option would pay a fixedamount of currency should the value of the underlying security, index orvariable be at or below the strike price of the put option. A spread ofeither digital call or put options would pay a fixed amount should theunderlying value expire at or between the strike prices. A strip ofdigital options would pay out fixed ratios should the underlying expirebetween two sets of strike prices. Graphically, digital calls, puts,spreads, and strips can have simple representations as shown in FIG. 26.As depicted in Graphs 6.0.1, 6.0.2, 6.0.3, and 6.0.4 shown in FIG. 26,the strike prices for the respective options are marked using familiaroptions notation where the subscript “c” indicates a call, the subscript“p” indicates a put, the subscript “s” indicates “spread,” and thesuperscripts “l” and “u” indicate lower and upper strikes, respectively.

A difference between digital options, which are frequently transacted inthe OTC foreign currency options markets, and traditional options suchas the equity options, which trade on the Chicago Board Options Exchange(“CBOE”), is that digital options have payouts which do not vary withthe extent to which the underlying asset, index, or variable(“underlying”) finishes in or out of the money. For example, a digitalcall option at a strike price for the underlying stock at 50 would paythe same amount if, at the fulfillment of all of the terminationcriteria, the underlying stock price was 51, 60, 75 or any other valueat or above 50. In this sense, digital options represent the academicfoundations of options theory, since traditional equity options could intheory be replicated from a portfolio of digital spread options whosestrike prices are set to provide vanishingly small spreads. (In fact, a“butterfly spread” of the traditional options yields a digital optionspread as the strike prices of the traditional options are allowed toconverge.) As can be seen from Graphs 6.0.1, 6.0.2, 6.0.3, and 6.0.4shown in FIG. 26, digital options can be constructed from digital optionspreads.

The methods and systems of the present invention can be used to create aderivatives market for digital options spreads. In other words, eachinvestment in a state of a mutually exclusive and collectivelyexhaustive set of states of a group of DBAR contingent claims can beconsidered to correspond to either a digital call spread or a digitalput spread. Since digital spreads can readily and accurately be used toreplicate digital options, and since digital options are known, tradedand processed in the existing markets, DBAR methods can therefore berepresented effectively as a market for digital options—that is, a DBARdigital options market.

6.1 Representation of Digital Options as DBAR Contingent Claims

One advantage of the digital options representation of DBAR contingentclaims is that the trader interface of a DBAR digital options exchange(a “DBAR DOE”) can be presented in a format familiar to traders, eventhough the underlying DBAR market structure is quite novel and differentfrom traditional securities and derivatives markets. For example, themain trader interface for a DBAR digital options exchange, in apreferred embodiment, could have the following features:

TABLE 6.1.1 MSFT Digital Options CALLS PUTS IND IND IND IND IND INDSTRIKE BID OFFER PAYOUT BID OFFER PAYOUT 30 0.9388 0.9407 1.0641 0.05930.0612 16.5999 40 0.7230 0.7244 1.3818 0.2756 0.2770 3.6190 50 0.43990.4408 2.2708 0.5592 0.5601 1.7869 60 0.2241 0.2245 4.4582 0.7755 0.77591.2892 70 0.1017 0.1019 9.8268 0.8981 0.8983 1.1133 80 0.0430 0.043123.2456 0.9569 0.9570 1.0450The illustrative interface of Table 6.1.1 contains hypothetical marketinformation on DBAR digital options on Microsoft stock (“MSFT”) for agiven expiration date. For example, an investor who desires a payout ifMSFT stock closes higher than 50 at the expiration or observation datewill need to “pay the offer” of $0.4408 per dollar of payout. Such anoffer is “indicative” (abbreviated “IND”) since the underlying DBARdistribution—that is, the implied probability that a state or set ofstates will occur—may change during the trading period. In a preferredembodiment, the bid/offer spreads presented in Table 6.1.1 are presentedin the following manner. The “offer” side in the market reflects theimplied probability that underlying value of the stock (in this exampleMSFT) will finish “in the money.” The “bid” side in the market is the“price” at which a claim can be “sold” including the transaction fee.(In this context, the term “sold” reflects the use of the systems andmethods of the present invention to implement investment profit and lossscenarios comparable to “sales” of digital options, discussed in detailbelow.) The amount in each “offer” cell is greater than the amount inthe corresponding “bid” cell. The bid/offer quotations for these digitaloption representations of DBAR contingent claims are presented aspercentages of (or implied probabilities for) a one dollar indicativepayout.

The illustrative quotations in Table 6.1.1 can be derived as follows.First the payout for a given investment is computed assuming a 10 basispoint transaction fee. This payout is equal to the sum of allinvestments less 10 basis points, divided by the sum of the investmentsover the range of states corresponding to the digital option. Taking theinverse of this quantity gives the offer side of the market in “price”terms. Performing the same calculation but this time adding 10 basispoints to the total investment gives the bid side of the market.

In another preferred embodiment, transaction fees are assessed as apercentage of payouts, rather than as a function of invested amounts.Thus, the offer (bid) side of the market for a given digital optioncould be, for example, (a) the amount invested over the range of statescomprising the digital option, (b) plus (minus) the fee (e.g., 10 basispoints) multiplied by the total invested for all of the defined states,(c) divided by the total invested for all of the defined states. Anadvantage of computing fees based upon the payout is that the bid/offerspreads as a percentage of “price” would be different depending upon thestrike price of the underlying, with strikes that are less likely to be“in the money” having a higher percentage fee. Other embodiments inwhich the exchange or transaction fees, for example, depend on the timeof trade to provide incentives for traders to trade early or to tradecertain strikes, or otherwise reflect liquidity conditions in thecontract, are apparent to those of skill in the art.

As explained in detail below, in preferred embodiments of the systemsand methods of the present invention, traders or investors cay buy and“sell” DBAR contingent claims that are represented and behave likedigital option puts, calls, spreads, and strips using conditional or“limit” orders. In addition, these digital options can be processedusing existing technological infrastructure in place at currentfinancial institutions. For example, Sungard, Inc., has a largesubscriber base to many off-the-shelf programs which are capable ofvaluing, measuring the risk, clearing, and settling digital options.Furthermore, some of the newer middleware protocols such as FINXML (seewww.finxml.org) apparently are able to handle digital options and otherswill probably follow shortly (e.g., FPML). In addition, the transactioncosts of a digital options exchange using the methods and systems of thepresent invention can be represented in a manner consistent with theconventional markets, i.e., in terms of bid/offer spreads.

6.2 Construction of Digital Options Using DBAR Methods and Systems

The methods of multistate trading of DBAR contingent claims previouslydisclosed can be used to implement investment in a group of DBARcontingent claims that behave like a digital option. In particular, andin a preferred embodiment, this can be accomplished by allocating aninvestment, using the multistate methods previously disclosed, in such amanner that the same payout is received from the investment should theoption expire “in-the-money”, e.g., above the strike price of theunderlying for a call option and below the strike price of theunderlying for a put. In a preferred embodiment, the multistate methodsused to allocate the investment need not be made apparent to traders. Insuch an embodiment, the DBAR methods and systems of the presentinvention could effectively operate “behind the scenes” to improve thequality of the market without materially changing interfaces and tradingscreens commonly used by traders. This may be illustrated by consideringthe DBAR construction of the MSFT Digital Options market activity asrepresented to the user in Table 6.1.1. For purposes of thisillustration, it is assumed that the market “prices” or impliedprobabilities for the digital put and call options as displayed in Table6.1.1 result from $100 million in investments. The DBAR states andallocated investments that construct these “prices” are then:

TABLE 6.2.1 States State Prob State Investments  (0, 30] 0.0602387$6,023,869.94 (30, 40] 0.2160676 $21,606,756.78 (40, 50] 0.2833203$28,332,029.61 (50, 60] 0.2160677 $21,606,766.30 (60, 70] 0.1225432$12,254,324.67 (70, 80] 0.0587436 $5,874,363.31 (80, ∞]  0.0430189$4,301,889.39In Table 6.2.1, the notation (x, y] is used to indicate a single statepart of a set of mutually exclusive and collectively exhaustive stateswhich excludes x and includes y on the interval.

(For purposes of this specification a convention is adopted for puts,calls, and spreads which is consistent with the internal representationof the states. For example, a put and a call both struck at 50 cannotboth be paid out if the underlying asset, index or variable expiresexactly at 50. To address this issue, the following convention could beadopted: calls exclude the strike price, puts include the strike price,and spreads exclude the lower and include the upper strike price. Thisconvention, for example, would be consistent with internal states thatare exclusive on the lower boundary and inclusive on the upper boundary.Another preferred convention would have calls including the strike priceand puts excluding the strike price, so that the representation of thestates would be inclusive on the lower boundary and exclusive on theupper. In any event, related conventions exist in traditional markets.For example, consider the situation of a traditional foreign exchangeoptions dealer who sells an “at the money” digital and an “at the money”put, with strike price of 100. Each is equally likely to expire “in themoney,” so for every $1.00 in payout, the dealer should collect $0.50.If the dealer has sold a $1.00 digital call and put, and has thereforecollected a total of $1.00 in premium, then if the underlying expiresexactly at 100, a discontinuous payout of $2.00 is owed. Hence, in apreferred embodiment of the present invention, conventions such as thosedescribed above or similar methods may be adopted to avoid suchdiscontinuities.)

A digital call or put may be constructed with DBAR methods of thepresent invention by using the multistate allocation algorithmspreviously disclosed. In a preferred embodiment, the construction of adigital option involves allocating the amount to be invested across theconstituent states over which the digital option is “in-the-money”(e.g., above the strike for a call, below the strike for a put) in amanner such that the same payout is obtained regardless of which stateoccurs among the “in the money” constituent states. This is accomplishedby allocating the amount invested in the digital option in proportion tothe then-existing investments over the range of constituent states forwhich the option is “in the money.” For example, for an additional$1,000,000 investment a digital call struck at 50 from the investmentsillustrated in Table 6.2.1, the construction of the trade usingmultistate allocation methods is:

TABLE 6.2.2 Internal States $1,000,000.00  (0, 30] (30, 40] (40, 50](50, 60] $490,646.45 (60, 70] $278,271.20 (70, 80] $133,395.04 (80, ∞]$97,687.30As other traders subsequently make investments, the distribution ofinvestments across the states comprising the digital option may change,and may therefore require that the multistate investments be reallocatedso that, for each digital option, the payout is the same for any of itsconstituent “in the money” states, regardless of which of theseconstituent states occurs after the fulfillment of all of thetermination criteria, and is zero for any of the other states. When theinvestments have been allocated or reallocated so that this payoutscenario occurs, the group of investments or contract is said to be inequilibrium. A further detailed description of the allocation methodswhich can be used to achieve this equilibrium is provided in connectionwith the description of FIGS. 13-14.

6.3 Digital Option Spreads

In a preferred embodiment, a digital option spread trade may be offeredto investors which simultaneously execute a buy and a “sell” (in thesynthetic or replicated sense of the term, as described below) of adigital call or put option. An investment in such a spread would havethe same payout should the underlying outcome expire at any valuebetween the lower and upper strike prices in the spread. If the spreadcovers one state, then the investment is comparable to an investment ina DBAR contingent claim for that one state. If the spread covers morethan one constituent state, in a preferred embodiment the investment isallocated using the multistate investment method previously described sothat, regardless of which state occurs among the states included in thespread trade, the investor receives the same payout.

6.4 Digital Option Strips

Traders in the derivatives markets commonly trade related groups offutures or options contracts in desired ratios in order to accomplishsome desired purpose. For example, it is not uncommon for traders ofLIBOR based interest rate futures on the Chicago Mercantile Exchange(“CME”) to execute simultaneously a group of futures with differentexpiration dates covering a number of years. Such a group, which iscommonly termed a “strip;” is typically traded to hedge another positionwhich can be effectively approximated with a strip whose constituentcontracts are executed in target relative ratios. For example, a stripof LIBOR-based interest rate futures may be used to approximate the riskinherent of an interest rate swap of the same maturity as the latestcontract expiration date in the strip.

In a preferred embodiment, the DBAR methods of the present invention canbe used to allow traders to construct strips of digital options anddigital option spreads whose relative payout ratios, should each optionexpire in the money, are equal to the ratios specified by the trader.For example, a trader may desire to invest in a strip consisting of the50, 60, 70, and 80 digital call options on MSFT, as illustrated in Table6.1.1. Furthermore, and again as an illustrative example, the trader maydesire that the payout ratios, should each option expire in the money,be in the following relative ratio: 1:2:3:4. Thus, should the underlyingprice of MSFT at the expiration date (when the event outcome isobserved) be equal to 65, both the 50 and 60 strike digital options arein the money. Since the trader desires that the 60 strike digital calloption pay out twice as much as the 50 strike digital call option, amultistate allocation algorithm, as previously disclosed and describedin detail, can be used dynamically to reallocate the trader'sinvestments across the states over which these options are in the money(50 and above, and 60 and above, respectively) in such a way as togenerate final payouts which conform to the indicated ratio of 1:2. Aspreviously disclosed, the multistate allocation steps may be performedeach time new investments are added during the trading period, and afinal multistate allocation may be performed after the trading periodhas expired.

6.5 Multistate Allocation Algorithm for Replicating “Sell” Trades

In a preferred embodiment of a digital options exchange using DBARmethods and systems of the present invention, traders are able to makeinvestments in DBAR contingent claims which correspond to purchases ofdigital options. Since DBAR methods are inherently demand-based—i.e., aDBAR exchange or market functions without traditional sellers—anadvantage of the multistate allocation methods of the present inventionis the ability to generate scenarios of profits and losses (“P&L”)comparable to the P&L scenarios obtained from selling digital options,spreads, and strips in traditional, non-DBAR markets without traditionalsellers or order-matching.

In traditional markets, the act of selling a digital option, spread, orstrip means that the investor (in the case of a sale, a seller) receivesthe cost of the option, or premium, if the option expires worthless orout of the money. Thus, if the option expires out of the money, theinvestor/seller's profit is the premium. Should the option expire in themoney, however, the investor/seller incurs a net liability equal to thedigital option payout less the premium received. In this situation, theinvestor/seller's net loss is the payout less the premium received forselling the option, or the notional payout less the premium. Selling anoption, which is equivalent in many respects to the activity of sellinginsurance, is potentially quite risky, given the large contingentliabilities potentially involved. Nonetheless, option selling iscommonplace in conventional, non-DBAR markets.

As indicated above, an advantage of the digital options representationof the DBAR methods of the present invention is the presentation of aninterface which displays bids and offers and therefore, by design,allows users to make investments in sets of DBAR contingent claims whoseP&L scenarios are comparable to those from traditional “sales” as wellas purchases of digital calls, puts, spreads, and strips. Specificallyin this context, “selling” entails the ability to achieve a profit andloss profile which is analogous to that achieved by sellers of digitaloptions instruments in non-DBAR markets, i.e., achieving a profit equalto the premium should the digital option expire out of the money, andsuffering a net loss equal to the digital option payout (or thenotional) less the premium received should the digital option expire inthe money.

In a preferred embodiment of a digital options exchange using the DBARcontingent claims methods and systems of the present invention, themechanics of “selling” involves converting such “sell” orders tocomplementary buy orders. Thus, a sale of the MSFT digital put optionswith strike price equal to 50, would be converted, in a preferred DBARDOE embodiment, to a complementary purchase of the 50 strike digitalcall options. A detailed explanation of the conversion process of a“sale” to a complementary buy order is provided in connection with thedescription of FIG. 15.

The complementary conversion of DBAR DOE “sales” to buys is facilitatedby interpreting the amount to be “sold” in a manner which is somewhatdifferent from the amount to be bought for a DBAR DOE buy order. In apreferred embodiment, when a trader specifies an amount in an order tobe “sold,” the amount is interpreted as the total amount of loss thatthe trader will suffer should the digital option, spread, or strip soldexpire in the money. As indicated above, the total amount lost or netloss is equal to the notional payout less the premium from the sale. Forexample, if the trader “sells” $1,000,000 of the MSFT digital put struckat 50, if the price of MSFT at expiration is 50 or below, then thetrader will lose $1,000,000. Correspondingly, in a preferred embodimentof the present invention, the order amount specified in a DBAR DOE“sell” order is interpreted as the net amount lost should the option,strip, or spread sold expire in the money. In conventional optionsmarkets, the amount would be interpreted and termed a “notional” or“notional amount” less the premium received, since the actual amountlost should the option expire in the money is the payout, or notional,less the premium received. By contrast, the amount of a buy order, in apreferred DBAR DOE embodiment, is interpreted as the amount to beinvested over the range of defined states which will generate the payoutshape or profile expected by the trader. The amount to be invested istherefore equivalent to the option “premium” in conventional optionsmarkets. Thus, in preferred embodiments of the present invention, forDBAR DOE buy orders, the order amount or premium is known and specifiedby the trader, and the contingent gain or payout should the optionpurchased finish in the money is not known until after all trading hasceased, the final equilibrium contingent claim “prices” or impliedprobabilities are calculated and any other termination criteria arefulfilled. By contrast, for a “sell” order in a preferred DBAR DOEembodiment of the present invention, the amount specified in the orderis the specified net loss (equal to the notional less the premium) whichrepresents the contingent loss should the option expire in the money.Thus, in a preferred embodiment, the amount of a buy order isinterpreted as an investment amount or premium which generates anuncertain payout until all predetermined termination criteria have beenmet; and the amount of a “sell” order is interpreted as a certain netloss should the option expire in the money corresponding to aninvestment amount or premium that remains uncertain until allpredetermined termination criteria have been met. In other words, in aDBAR DOE preferred embodiment, buy orders are for “premium” while “sell”orders are for net loss should the option expire in the money.

A relatively simple example illustrates the process, in a preferredembodiment of the present invention, of converting a “sale” of a DBARdigital option, strip, or spread to a complementary buy and the meaningof interpreting the amount of a buy order and “sell” order differently.Referring the MSFT example illustrated in Table 6.1.1 and Table 6.2.1above, assume that a trader has placed a market order (conditional orlimit orders are described in detail below) to “sell” the digital putwith strike price equal to 50. Ignoring transaction costs, the “price”of the 50 digital put option is equal to the sum of the implied stateprobabilities spanning the states where the option is in the money(i.e., (0,30),(30,40], and (40,50]) and is approximately 0.5596266. Whenthe 50 put is in the money, the 50 call is out of the money and viceversa. Accordingly, the 50 digital call is “complementary” to the 50digital put. Thus, “selling” the 50 digital put for a given amount isequivalent in a preferred embodiment to investing that amount in thecomplementary call, and that amount is the net loss that would besuffered should the 50 digital put expire in the money (i.e., 50 andbelow). For example, if a trader places a market order to “sell”1,000,000 value units of the 50 strike digital put, this 1,000,000 valueunits are interpreted as the net loss if the digital put option expiresin the money, i.e., it corresponds to the notional payout loss plus thepremium received from the “sale.”

In preferred embodiments of the present investment, the 1,000,000 valueunits to be “sold” are treated as invested in the complementary50-strike digital call, and therefore are allocated according to themultistate allocation algorithm described in connection with thedescription of FIG. 13. The 1,000,000 value units are allocated inproportion to the value units previously allocated to the range ofstates comprising the 50-strike digital call, as indicated in Table6.2.2 above. Should the digital put expire in the money, the trader“selling” the digital put loses 1,000,000 value units, i.e., the traderloses the payout or notional less the premium. Should the digital putfinish out of the money, the trader will receive a payout approximatelyequal to 2,242,583.42 value units (computed by taking the total amountof value units invested, or 101,000,000, dividing by the new totalinvested in each state above 50 where the digital put is out of themoney, and multiplying by the corresponding state investment). Thepayout is the same regardless of which state above 50 occurs uponfulfillment of the termination criteria, i.e., the multistate allocationhas achieved the desired payout profile for a digital option. In thisillustration, the “sell” of the put will profit by 1,242,583.42 shouldthe option sold expire out of the money. This profit is equivalent tothe premium “sold.” On the other hand, to achieve a net loss of1,000,000 value units from a payout of 2,242,583.42, the premium is setat 1,242,583.42 value units.

The trader who “sells” in a preferred embodiment of a DBAR DOE specifiesan amount that is the payout or notional to be sold less the premium tobe received, and not the profit or premium to be made should the optionexpire out of the money. By specifying the payout or notional “sold”less the premium, this amount can be used directly as the amount to beinvested in the complementary option, strip, or spread. Thus, in apreferred embodiment, a DBAR digital options exchange can replicate orsynthesize the equivalent of trades involving the sale of option payoutsor notional (less the premium received) in the traditional market.

In another preferred embodiment, an investor may be able to specify theamount of premium to be “sold.” To illustrate this embodiment, quantityof premium to be “sold” can be assigned to the variable x. An investmentof quantity y on the states complementary to the range of states being“sold” is related to the premium x in the following manner:

${\frac{y}{1 - p} - y} = x$

where p is the final equilibrium “price”, including the “sale” x (andthe complementary investment y) of the option being “sold.” Rearrangingthis expression yields the amount of the complementary buy investment ythat must be made to effect the “sale” of the premium x:

$y = {x*\frac{\left( {1 - p} \right)}{p}}$

From this it can be seen that, given an amount of premium x that isdesired to be “sold,” the complementary investment that must be boughton the complement states in order for the trader to receive the premiumx, should the option “sold” expire out of the money, is a function ofthe price of the option being “sold.” Since the price of the optionbeing “sold” can be expected to vary during the trading period, in apreferred embodiment of a DBAR DOE of the present invention, the amounty required to be invested in the complementary state as a buy order canalso be expected to vary during the trading period.

In a preferred embodiment, traders may specify an amount of notionalless the premium to be “sold” as denoted by the variable y. Traders maythen specify a limit order “price” (see Section 6.8 below for discussionof limit orders) such that, by the previous equation relating y to x, atrader may indirectly specify a minimum value of x with the specifiedlimit order “price,” which may be substituted for p in the precedingequation. In another preferred embodiment, an order containingiteratively revised y amounts, as “prices” change during the tradingperiod are submitted. In another preferred embodiment, recalculation ofequilibrium “prices” with these revised y amounts is likely to lead to aconvergence of the y amounts in equilibrium. In this embodiment aniterative procedure may be employed to seek out the complementary buyamounts that must be invested on the option, strip, or spreadcomplementary to the range of states comprising the option being “sold”in order to replicate the desired premium that the trader desired to“sell.” This embodiment is useful since it aims to make the act of“selling” in a DBAR DOE more similar to the traditional derivativesmarkets.

It should be emphasized that the traditional markets differ from thesystems and methods of the present invention in as least one fundamentalrespect. In traditional markets, the sale of an option requires a sellerwho is willing to sell the option at an agreed-upon price. An exchangeof DBAR contingent claims of the present invention, in contrast, doesnot require or involve such sellers. Rather, appropriate investments maybe made (or bought) in contingent claims in appropriate states so thatthe payout to the investor is the same as if the claim, in a traditionalmarket, had been sold. In particular, using the methods and systems ofthe present invention, the amounts to be invested in various states canbe calculated so that the payout profile replicates the payout profileof a sale of a digital option in a traditional market, but without theneed for a seller. These steps are described in detail in connectionwith FIG. 15.

6.6 Clearing and Settlement

In a preferred embodiment of a digital options exchange using the DBARcontingent claims systems and methods of the present invention, alltypes of positions may be processed as digital options. This is becauseat fixing (i.e., the finalization of contingent claim “prices” orimplied probabilities at the termination of the trading period or otherfulfillment of all of the termination criteria) the profit and lossexpectations of all positions in the DBAR exchange are, from thetrader's perspective, comparable to if not the same as the profit andloss expectations of standard digital options commonly traded in the OTCmarkets, such as the foreign exchange options market (but without thepresence of actual sellers, who are needed on traditional optionsexchanges or in traditional OTC derivatives markets). The contingentclaims in a DBAR DOE of the present invention, once finalized at the endof a trading period, may therefore be processed as digital options orcombinations of digital options. For example, a MSFT digital option callspread with a lower strike of 40 and upper strike of 60 could beprocessed as a purchase of the lower strike digital option and a sale ofthe upper strike digital option.

There are many vendors of back office software that can readily handlethe processing of digital options. For example, Sungard, Inc., producesa variety of mature software systems for the processing of derivativessecurities, including digital options. Furthermore, in-house derivativessystems currently in use at major banks have basic digital optionscapability. Since digital options are commonly encountered instruments,many of the middleware initiatives currently underway e.g., FINXML, willlikely incorporate a standard protocol for handling digital options.Therefore, an advantage of a preferred embodiment of the DBAR DOE of thepresent invention is the ability to integrate with and otherwise useexisting technology for such an exchange.

6.7 Contract Initialization

Another advantage of the systems and methods of the present invention isthat, as previously noted, digital options positions can be representedinternally as composite trades. Composite trades are useful since theyhelp assure that an equilibrium distribution of investments among thestates can be achieved. In preferred embodiments, digital option andspreading activity will contribute to an equilibrium distribution. Thus,in preferred embodiments, indicative distributions may be used toinitialize trading at the beginning of the trading period.

In a preferred embodiment, these initial distributions may berepresented as investments in each of the defined states making up thecontract or group of DBAR contingent claims. Since these investmentsneed not be actual trader investments, they may be reallocated among thedefined states as actual trading occurs, so long as the initialinvestments do not change the implicit probabilities of the statesresulting from actual investments. In a preferred embodiment, thereallocation of initial investments is performed gradually so as tomaximize the stability of digital call and put “prices” (and spreads),as viewed by investors. By the end of the trading period, all of theinitial investments may be reallocated in proportion to the investmentsin each of the defined states made by actual traders. The reallocationprocess may be represented as a composite trade that has a same payoutirrespective of which of the defined states occurs. In preferredembodiments the initial distribution can be chosen using current marketindications from the traditional markets to provide guidance fortraders, e.g., options prices from traditional option markets can beused to calculate a traditional market consensus probabilitydistribution, using for example, the well-known technique of Breeden andLitzenberger. Other reasonable initial and indicative distributionscould be used. Alternatively, in a preferred embodiment, initializationcan be performed in such a manner that each defined state is initializedwith a very small amount, distributed equally among each of the definedstates. For example, each of the defined states could be initializedwith 10⁻⁶ value units. Initialization in this manner is designed tostart each state with a quantity that is very small, distributed so asto provide a very small amount of information regarding the impliedprobability of each defined state. Other initialization methods of thedefined states are possible and could be implemented by one of skill inthe art.

6.8 Conditional Investments, or Limit Orders

In a preferred embodiment of the system and methods of the presentinvention, traders may be able to make investments which are onlybinding if a certain “price” or implied probability for a given state ordigital option (or strip, spread, etc.) is achieved. In this context,the word “price,” is used for convenience and familiarity and, in thesystems and methods of the present invention, reflects the impliedprobability of the occurrence of the set of states corresponding to anoption—i.e., the implied probability that the option expires “in themoney.” For instance, in the example reflected in Table 6.2.1, a tradermay wish to make an investment in the MSFT digital call options withstrike price of 50, but may desire that such an investment actually bemade only if the final equilibrium “price” or implied probability is0.42 or less. Such a conditional investment, which is conditional uponthe final equilibrium “price” for the digital option, is sometimesreferred to (in conventional markets) as a “limit order.” Limit ordersare popular in traditional markets since they provide the means forinvestors to execute a trade at “their price” or better. Of course,there is no guarantee that such a limit order—which may be placedsignificantly away from the current market price—will in fact beexecuted. Thus, in traditional markets, limit orders provide the meansto control the price at which a trade is executed, without the traderhaving to monitor the market continuously. In the systems and method ofthe present invention, limit orders provide a way for investors tocontrol the likelihood that their orders will be executed at theirpreferred “prices” (or better), also without having continuously tomonitor the market.

In a preferred embodiment of a DBAR DOE, traders are permitted to buyand sell digital call and put options, digital spreads, and digitalstrips with limit “prices” attached. The limit “price” indicates that atrader desires that his trade be executed at that indicated limit“price”—actually the implied probability that the option will expire inthe money—“or better.” In the case of a purchase of a digital option,“better” means at the indicated limit “price” implied probability orlower (i.e., purchasing not higher than the indicated limit “price”). Inthe case of a “sale” of a DBAR digital option, “better” means at theindicated limit “price” (implied probability) or higher (i.e., sellingnot lower than the indicated limit “price”).

A benefit of a preferred embodiment of a DBAR DOE of the presentinvention which includes conditional investments or limit orders is thatthe placing of limit orders is a well-known mechanism in the financialmarkets. By allowing traders and investors to interact with a DBAR DOEof the present invention using limit orders, more liquidity should flowinto the DBAR DOE because of the familiarity of the mechanism, eventhough the underlying architecture of the DBAR DOE is different from theunderlying architecture of other financial markets.

The present invention also includes novel methods and systems forcomputing the equilibrium “prices” or implied probabilities, in thepresence of limit orders, of DBAR contingent claims in the variousstates. These methods and systems can be used to arrive at anequilibrium exclusively in the presence of limit orders, exclusively inthe presence of market orders, and in the presence of both. In apreferred embodiment, the steps to compute a DBAR DOE equilibrium for agroup of contingent claims including at least one limit order aresummarized as follows:

-   -   6.8(1) Convert all “sale” orders to complementary buy orders.        This is achieved by (i) identifying the states complementary to        the states being sold; (ii) using the amount “sold” as the        amount to be invested in the complementary states, and;        and (iii) for limit orders, adjusting the limit “price” to one        minus the original limit “price.”    -   6.8(2) Group the limit orders by placing all of the limit orders        which span or comprise the same range of defined states into the        same group. Sort each group from the best (highest “price” buy)        to the worst (lowest “price” buy). All orders may be processed        as buys since any “sales” have previously been converted to        complementary buys. For example, in the context of the MSFT        Digital Options illustrated in Table 6.2.1, there would be        separate groups for the 30 digital calls, the 30 digital puts,        the 40 digital calls, the 40 digital puts, etc. In addition,        separate groups are made for each spread or strip that spans or        comprises a distinct set of defined states.    -   6.8(3) Initialize the contract or group of DBAR contingent        claim. This may be done, in a preferred embodiment, by        allocating minimal quantities of value units uniformly across        the entire distribution of defined states so that each defined        state has a non-zero quantity of value units.    -   6.8(4) For all limit orders, adjust the limit “prices” of such        orders by subtracting from each limit order the order,        transaction or exchange fees for the respective contingent        claims.    -   6.8(5) With all orders broken into minimal size unit lots (e.g.,        one dollar or other small value unit for the group of DBAR        contingent claims), identify one order from a group that has a        limit “price” better than the current equilibrium “price” for        the option, spread, or strip specified in the order.    -   6.8(6) With the identified order, find the maximum number of        additional unit lots (“lots”) than can be invested such that the        limit “price” is no worse than the equilibrium “price” with the        chosen maximum number of unit lots added. The maximum number of        lots can be found by (i) using the method of binary search, as        described in detail below, (ii) trial addition of those lots to        already-invested amounts and (iii) recalculating the equilibrium        iteratively.    -   6.8(7) Identify any orders which have limit “prices” worse than        the current calculated equilibrium “prices” for the contract or        group of DBAR contingent claims. Pick such an order with the        worst limit “price” from the group containing the order. Remove        the minimum quantity of unit lots required so that the order's        limit “price” is no longer worse than the equilibrium “price”        calculated when the unit lots are removed. The number of lots to        be removed can be found by (i) using the method of binary        search, as described in detail below, (ii) trial subtraction of        those lots from already invested amounts and (iii) recalculating        the equilibrium iteratively.    -   6.8(8) Repeat steps 6.8(5) to 6.8(7). Terminate those steps when        no further additions or removals are necessary.    -   6.8(9) Optionally, publish the equilibrium from step 6.8(8) both        during the trading period and the final equilibrium at the end        of the trading period. The calculation during the trading period        is performed “as if” the trading period were to end at the        moment the calculation is performed. All prices resulting from        the equilibrium computation are considered mid-market prices,        i.e., they do not include the bid and offer spreads owing to        transaction fees. Published offer (bid) “prices” are set equal        to the mid-market equilibrium “prices” plus (minus) the fee.

In a preferred embodiment, the preceding steps 6.8(1) to 6.8(8) andoptionally step 6.8(9) are performed each time the set of orders duringthe trading or auction period changes. For example, when a new order issubmitted or an existing order is cancelled (or otherwise modified) theset of orders changes, steps 6.8(1) to 6.8(8) (and optionally step6.8(9)) would need to be repeated.

The preceding steps result in an equilibrium of the DBAR contingentclaims and executable orders which satisfy typical trader expectationsfor a market for digital options:

-   -   (1) At least some buy (“sell”) orders with a limit “price”        greater (less) than or equal to the equilibrium “price” for the        given option, spread or strip are executed or “filled.”    -   (2) No buy (“sell”) orders with limit “prices” less (greater)        than the equilibrium “price” for the given option, spread or        strip are executed.    -   (3) The total amount of executed lots equals the total amount        invested across the distribution of defined states.    -   (4) The ratio of payouts should each constituent state of a        given option, spread, or strike occur is as specified by the        trader, (including equal payouts in the case of digital        options), within a tolerable degree of deviation.    -   (5) Conversion of filled limit orders to market orders for the        respective filled quantities and recalculating the equilibrium        does not materially change the equilibrium.    -   (6) Adding one or more lots to any of the filled limit orders        converted to market orders in step (5) and recalculating of the        equilibrium “prices” results in “prices” which violate the limit        “price” of the order to which the lot was added (i.e., no more        lots can be “squeaked in” without forcing market prices to go        above the limit “prices” of buy orders or below the limit        “prices” of sell orders).

The following example illustrates the operation of a preferredembodiment of a DBAR DOE of the present invention exclusively with limitorders. It is anticipated that a DBAR DOE will operate and process bothlimit and non-limit or market orders. As apparent to a person of skillin the art, if a DBAR DOE can operate with only limit orders, it canalso operate with both limit orders and market orders.

Like earlier examples, this example is also based on digital optionsderived from the price of MSFT stock. To reduce the complexity of theexample, it is assumed, for purposes of illustration, that there areillustrative purposes, only three strike prices: $30, $50, and $80.

TABLE 6.8.1 Buy Orders Limit Limit “Price” Quantity “Price” QuantityLimit “Price” Quantity 30 calls 50 calls 80 calls 0.82 10000 0.43 100000.1 10000 0.835 10000 0.47 10000 0.14 10000 0.84 10000 0.5 10000 80 puts50 puts 30 puts 0.88 10000 0.5 10000 0.16 10000 0.9 10000 0.52 100000.17 10000 0.92 10000 0.54 10000

TABLE 6.8.2 “Sell” Orders Limit Limit “Price” Quantity “Price” QuantityLimit “Price” Quantity 30 calls 50 calls 80 calls 0.81  5000 0.42 100000.11 10000 0.44 10000 0.12 10000 80 puts 50 puts 30 puts 0.9 20000 0.4510000 0.15  5000 0.50 10000 0.16 10000The quantities entered in the “Sell Orders” table, Table 6.8.2, are thenet loss amounts which the trader is risking should the option “sold”expire in the money, i.e., they are equal to the notional less thepremium received from the sale, as discussed above.

-   -   (i) According to step 6.8(1) of the limit order methodology        described above, the “sale” orders are first converted to buy        orders. This involves switching the contingent claim “sold” to a        buy of the complementary contingent claim and creating a new        limit “price” for the converted order equal to one minus the        limit “price” of the sale. Converting the “sell” orders in Table        6.8.2 therefore yields the following converted buy orders:

TABLE 6.8.3 “Sale” Orders Converted to Buy Orders Limit Limit “Price”Quantity “Price” Quantity Limit “Price” Quantity 30 puts 50 puts 80 puts0.19  5000 0.58 10000 0.89 10000 0.56 10000 0.88 10000 80 calls 50 calls30 calls 0.1 20000 0.55 10000 0.85  5000 0.50 10000 0.84 10000

-   -   (ii) According to step 6.8(2), the orders are then placed into        groupings based upon the range of states which each underlying        digital option comprises or spans. The groupings for this        illustration therefore are: 30 calls, 50 calls, 80 calls, 30        puts, 50 puts, and 80 puts    -   (iii) In this illustrative example, the initial liquidity in        each of the defined states is set at one value unit.    -   (iv) According to step 6.8(4), the orders are arranged from        worst “price” (lowest for buys) to best “price” (highest for        buys). Then, the limit “prices” are adjusted for the effect of        transaction or exchange costs. Assuming that the transaction fee        for each order is 5 basis points (0.0005 value units), then        0.0005 is subtracted from each limit order price. The aggregated        groups for this illustrative example, sorted by adjusted limit        prices (but without including the initial one-value-unit        investments), are as displayed in the following table:

TABLE 6.8.4 Aggregated, Sorted, Converted, and Adjusted Limit OrdersLimit Limit “Price” Quantity “Price” Quantity Limit “Price” Quantity 30calls 50 calls 80 calls 0.8495  5000 0.5495 10000 0.1395 10000 0.839520000 0.4995 20000 0.0995 30000 0.8345 10000 0.4695 10000 0.8195 100000.4295 10000 80 puts 50 puts 30 puts 0.9195 10000 0.5795 10000 0.1895 5000 0.8995 10000 0.5595 10000 0.1695 10000 0.8895 10000 0.5395 100000.1595 10000 0.8795 20000 0.5195 10000 0.4995 10000

-   -   -   After adding the initial liquidity of one value unit in each            state, the initial option prices are as follows:

TABLE 6.8.5 MSFT Digital Options Initial Prices CALLS PUTS IND IND INDIND IND IND STRIKE MID BID OFFER MID BID OFFER 30 0.85714 0.856640.85764 0.14286 0.14236 0.14336 50 0.57143 0.57093 0.57193 0.428570.42807 0.42907 80 0.14286 0.14236 0.14336 0.85714 0.85664 0.85764

-   -   (v) According to step 6.8(5) and based upon the description of        limit order processing in connection with FIG. 12, in this        illustrative example an order from Table 6.8.4 is identified        which has a limit “price” better or higher than the current        market “price” for a given contingent claim. For example, from        Table 6.9.4, there is an order for 10000 digital puts struck at        80 with limit “price” equal to 0.9195. The current mid-market        “price” for such puts is equal to 0.85714.    -   (vi) According to step 6.8(6), by the methods described in        connection with FIG. 17, the maximum number of lots of the order        for the 80 digital puts is added to already-invested amounts        without increasing the recalculated mid-market “price,” with the        added lots, above the limit order price of 0.9195. This process        discovers that, when five lots of the 80 digital put order for        10000 lots and limit “price” of 0.9195 are added, the new        mid-market price is equal to 0.916667. Assuming the distribution        of investments for this illustrative example, addition of any        more lots will drive the mid-market price above the limit price.        With the addition of these lots, the new market prices are:

TABLE 6.8.5 MSFT Digital Options Prices after addition of five lots of80 puts CALLS PUTS IND IND IND IND IND IND STRIKE MID BID OFFER MID BIDOFFER 30 0.84722 0.84672 0.84772 0.15278 0.15228 0.15328 50 0.541670.54117 0.54217 0.45833 0.45783 0.45883 80 0.08333 0.08283 0.083830.91667 0.91617 0.91717

-   -   -   As can be seen from Table 6.8.5, the “prices” of the call            options have decreased while the “prices” of the put options            have increased as a result of filling five lots of the 80            digital put options, as expected.

    -   (vii) According to step 6.8(7), the next step is to determine,        as described in FIG. 17, whether there are any limit orders        which have previously been filled whose limit “prices” are now        less than the current mid-market “prices,” and as such, should        be subtracted. Since there are no orders than have been filled        other than the just filled 80 digital put, there is no removal        or “prune” step required at this stage in the process.

    -   (viii) According to step 6.8(8), the next step is to identify        another order which has a limit “price” higher than the current        mid-market “prices” as a candidate for lot addition. Such a        candidate is the order for 10000 lots of the 50 digital puts        with limit price equal to 0.5795. Again the method of binary        search is used to determine the maximum number of lots that can        be added from this order to already-invested amounts without        letting the recalculated mid-market “price” exceed the order's        limit price of 0.5795. Using this method, it can be determined        that only one lot can be added without forcing the new market        “price” including the additional lot above 0.5795. The new        prices with this additional lot are then:

TABLE 6.8.6 MSFT Digital Options “Prices” after (i) addition of fivelots of 80 puts and (ii) addition of one lot of 50 puts CALLS PUTS INDIND IND IND IND IND STRIKE MID BID OFFER MID BID OFFER 30 0.824200.82370 0.82470 0.17580 0.17530 0.17630 50 0.47259 0.47209 0.473090.52741 0.52691 0.52791 80 0.07692 0.07642 0.07742 0.923077 0.922580.92358

-   -   -   Continuing with step 6.8(8), the next step is to identify an            order whose limit “price” is now worse (i.e., lower than)            the mid-market “prices” from the most recent equilibrium            calculation as shown in Table 6.8.6. As can be seen from the            table, the mid-market “price” of the 80 digital put options            is now 0.923077. The best limit order (highest “priced”) is            the order for 10000 lots at 0.9195, of which five are            currently filled. Thus, the binary search routine determines            the minimum number of lots which are to be removed from this            order so that the order's limit “price” is no longer worse            (i.e., lower than) the newly recalculated market “price.”            This is the removal or prune part of the equilibrium            calculation.        -   The “add and prune” steps are repeated iteratively with            intermediate multistate equilibrium allocations performed.            The contract is at equilibrium when no further lots may be            added for orders with limit order “prices” better than the            market or removed for limit orders with “prices” worse than            the market. At this point, the group of DBAR contingent            claims (sometimes referred to as the “contract”) is in            equilibrium, which means that all of the remaining            conditional investments or limit orders—i.e., those that did            not get removed—receive “prices” in equilibrium which are            equal to or better than the limit “price” conditions            specified in each order. In the present illustration, the            final equilibrium “prices” are:

TABLE 6.8.7 MSFT Digital Options Equilibrium Prices CALLS PUTS IND INDIND IND IND IND STRIKE MID BID OFFER MID BID OFFER 30 0.830503 0.8300030.831003 0.169497 0.168997 0.169997 50 0.480504 0.480004 0.4810040.519496 0.518996 0.519996 80 0.139493 0.138993 0.139993 0.8605070.860007 0.861007

-   -   -   Thus, at these equilibrium “prices,” the following table            shows which of the original orders are executed or “filled”:

TABLE 6.8.8 Filled Buy Orders Limit “Price” Quantity Filled Limit“Price” Quantity Filled Limit “Price” Quantity Filled 30 calls 50 calls80 calls 0.82 10000 0 0.43 10000 0 0.1 10000 0 0.835 10000 10000 0.4710000 0 0.14 10000 8104 0.84 10000 10000 0.5 10000 10000 80 puts 50 puts30 puts 0.88 10000 10000 0.5 10000 0 0.16 10000 0 0.9 10000 10000 0.5210000 2425 0.17 10000 2148 0.92 10000 10000 0.54 10000 10000

TABLE 6.8.9 Filled Sell Orders Limit “Price” Quantity Filled Limit“Price” Quantity Filled Limit “Price” Quantity Filled 30 calls 50 calls80 calls 0.81 5000 5000 0.42 10000 10000 0.11 10000 10000 0.44 1000010000 0.12 10000 10000 80 puts 50 puts 30 puts 0.9 20000 0 0.45 1000010000 0.15 5000 5000 0.50 10000 10000 0.16 10000 10000

It may be possible only partially to execute or “fill” a trader's orderat a given limit “price” or implied probability of the relevant states.For example, in the current illustration, the limit buy order for 50puts at limit “price” equal to 0.52 for an order amount of 10000 may beonly filled in the amount 2424 (see Table 6.8.8). If orders are made bymore than one investor and not all of them can be filled or executed ata given equilibrium, in preferred embodiments it is necessary to decidehow many of which investor's orders can be filled, and how many of whichinvestor's orders will remain unfulfilled at that equilibrium. This maybe accomplished in several ways, including by filling orders on afirst-come-first-filled basis, or on a pro rata or other basis known orapparent to one of skill in the art. In preferred embodiments, investorsare notified prior to the commencement of a trading period about thebasis on which orders are filled when all investors' limit orders cannotbe filled at a particular equilibrium.

6.9 Sensitivity Analysis and Depth of Limit Order Book

In preferred embodiments of the present invention, traders in DBARdigital options may be provided with information regarding the quantityof a trade that could be executed (“filled”) at a given limit “price” orimplied probability for a given option, spread or strip. For example,consider the MSFT digital call option with strike of 50 illustrated inTable 6.1.1 above. Assume the current “price” or implied probability ofthe call option is 0.4408 on the “offer” side of the market. A tradermay desire, for example, to know what quantity of value units could betransacted and executed at any given moment for a limit “price” which isbetter than the market. In a more specific example, for a purchase ofthe 50 strike call option, a trader may want to know how much would befilled at that moment were the trader to specify a limit “price” orimplied probably of, for example, 0.46. This information is notnecessarily readily apparent, since the acceptance of conditionalinvestments (i.e., the execution of limit orders) changes the impliedprobability or “price” of each of the states in the group. As the limit“price” is increased, the quantities specified in a buy order are morelikely to be filled, and a curve can be drawn with the associated limit“price”/quantity pairs. The curve represents the amount that could befilled (for example, along the X-axis) versus the corresponding limit“price” or implied probability of the strike of the order (for example,along the Y-axis). Such a curve should be useful to traders, since itprovides an indication of the “depth” of the DBAR DOE for a givencontract or group of contingent claims. In other words, the curveprovides information on the “price” or implied probability, for example,that a buyer would be required to accept in order to execute apredetermined or specified number of value units of investment for thedigital option.

6.10 Networking of DBAR Digital Options Exchanges

In preferred embodiments, one or more operators of two or more differentDBAR Digital Options Exchanges may synchronize the time at which tradingperiods are conducted (e.g., agreeing on the same commencement andpredetermined termination criteria) and the strike prices offered for agiven underlying event to be observed at an agreed upon time. Eachoperator could therefore be positioned to offer the same trading periodon the same underlying DBAR event of economic significance or financialinstrument. Such synchronization would allow for the aggregation ofliquidity of two or more different exchanges by means of computing DBARDOE equilibria for the combined set of orders on the participatingexchanges. This aggregation of liquidity is designed to result in moreefficient “pricing” so that implied probabilities of the various statesreflect greater information about investor expectations than if a singleexchange were used.

7. DBAR DOE: Another Embodiment

In another embodiment of a DBAR Digital Options Exchange (“DBAR DOE”), atype of demand-based market or auction, all orders for digital optionsare expressed in terms of the payout (or “notional payout”) receivedshould any state of the set of constituent states of a DBAR digitaloption occur (as opposed to, for example, expressing buy digital optionorders in terms of premium to be invested and expressing “sell” digitaloption orders in terms of notional payout, or notional payout less thepremium received). In this embodiment, the DBAR DOE can accept andprocess limit orders for digital options expressed in terms of eachtrader's desired payout. In this embodiment, both buy and sell ordersmay be handled consistently, and the speed of calculation of theequilibrium calculation is increased. This embodiment of the DBAR DOEcan be used with or without limit orders (also referred to asconditional investments). Additionally this embodiment of the DBAR DOEcan be used to trade in a demand-based market or auction based on anyevent, regardless of whether the event is economically significant ornot.

In this embodiment, an equilibrium algorithm (set forth in Equations7.3.7 and 7.4.7) may be used on orders without limits (without limits onthe price), to determine the prices and total premium invested into aDBAR DOE market or auction based only upon information concerning therequested payouts per order and the defined states (or spreads) forwhich the desired digital option is in-the-money (the payout profile forthe order). The requested payout per order is the executed notionalpayout per order, and the trader or user pays the price determined atthe end of the trading period by the equilibrium algorithm necessary toreceive the requested payout.

In this embodiment, an optimization system (also referred to as theOrder Price Function or OPF) may also be utilized that maximizes thepayouts per order within the constraints of the limit order. In otherwords, when a user or trader specifies a limit order price, and alsospecifies the requested payouts per order and the defined states (orspreads) for which the desired digital option is in-the-money, then theoptimization system or OPF determines a price of each order that is lessthan or equal to each order's limit price, while maximizing the executednotional payout for the orders. As set forth below, in this limit orderexample, the user may not receive the requested payout but will receivea maximum executed notional payout given the limit price that the userdesires to invest for the payout.

In other words, in this embodiment, three mathematical principlesunderlie demand-based markets or auctions: demand-based pricing andself-funding conditions; how orders in digital options are constitutedin a demand-based market or auction; and, how a demand-based auction ormarket may be implemented with standard limit orders.

In this embodiment, for each demand-based market or auction, thedemand-based pricing condition applies to every pair of fundamentalcontingent claims. In demand-based systems, the ratio of prices of eachpair of fundamental contingent claims is equal to the ratio of volumefilled for those claims. This is a notable feature of DBAR contingentclaims markets because the demand-based pricing condition relates theamount of relative volumes that may clear in equilibrium to the relativeequilibrium market prices. Thus, a demand-based market microstructure,which is the foundation of demand-based market or auction, is uniqueamong market mechanisms in that the relative prices of claims aredirectly related to the relative volume transacted of those claims. Bycontrast, in conventional markets, which have heretofore not adopteddemand-based principles, relative contingent claim prices typicallyreflect, in theory, the absence of arbitrage opportunities between suchclaims, but nothing is implied or can be inferred about the relativevolumes demanded of such claims in equilibrium.

Equation 7.4.7, as set forth below, is the equilibrium equation fordemand-based trading in accordance with one embodiment of the presentinvention. It states that a demand-based trading equilibrium can bemathematically expressed in terms of a matrix eigensystem, in which thetotal premium collected in a demand-based market or auction (T) is equalto the maximum eigenvalue of a matrix (H) which is a function of theaggregate notional amounts executed for each fundamental spread and theopening orders. In addition, the eigenvector corresponding to thismaximum eigenvalue, when normalized, contains the prices of thefundamental single strike spreads. Equation 7.4.7 shows that givenaggregate notional amounts to be executed (Y) and arbitrary amounts ofopening orders (K), that a unique demand-based trading equilibriumresults. The equilibrium is unique because a unique total premiuminvestment, T, is associated with a unique vector of equilibrium prices,p, by the solution of the eigensystem of Equation 7.4.7.

Demand-based markets or auctions may be implemented with a standardlimit order book in which traders attach price conditions for executionof buy and sell orders. As in any other market, limit orders allowtraders to control the price at which their orders are executed, at therisk that the orders may not be executed in full or in part. Limitorders may be an important execution control feature in demand-basedauctions or markets because final execution is delayed until the end ofthe trading or auction period.

Demand-based markets or auctions may incorporate standard limit ordersand limit order book principles. In fact, the limit order book employedin a demand-based market or auction and the mathematical expressionsused therein may be compatible with standard limit order book mechanismsfor other existing markets and auctions. The mathematical expression ofa General Limit Order Book is an optimization problem in which themarket clearing solution to the problem maximizes the volume of executedorders subject to two constraints for each order in the book. Accordingto the first constraint, should an order be executed, the order's limitprice is greater than or equal to the market price including theexecuted order. According to the second constraint, the order's executednotional amount is not to exceed the notional amount requested by thetrader to be executed.

7.1 Special Notation

For the purposes of the discussion of the embodiment described in thepresent section, the following notation is utilized. The notation usessome symbols previously employed in other sections of thisspecification. It should be understood that the meanings of thesenotational symbols are valid as defined below only in the context of thediscussion in the present section (Section 7—DBAR DOE: ANOTHEREMBODIMENT as well as the discussion in relation to FIG. 19 and FIG. 20in Section 9).

Known Variables

-   m: number of defined states or spreads, a natural number. Index    letter I, i=1, 2, . . . , m.-   k: m×1 vector where k_(i) is the initial invested premium for state    I, i=1, 2, . . . , m.    -   k_(i) is a natural number so k_(i)>0 i=1, 2, . . . , m-   e: a vector of ones of length m (m×1 unit vector)-   n: number of orders in the market or auction, a natural number.    Index letter j, j=1, 2, . . . , n-   r: n×1 vector where r_(j) is equal to the requested payout for order    j,    -   j=1, 2, . . . , n, r_(j) is a natural number so r_(j) is        positive for all j, j=1, 2, . . . , n-   w: n×1 vector where w_(j) equals the inputted limit price for order    j, j=1, 2, . . . , n    -   Range: 0<w_(j)≦1 for j=1, 2, . . . , n for digital options        -   0<w_(j) for j=1, 2, . . . , n for arbitrary payout options-   w_(j) ^(a): n×1 vector where w_(j) ^(a) is the adjusted limit price    for order j after converting “sell” orders into buy orders (as    discussed below) and after adjusting the inputted limit order w_(j)    with fee f_(j) (assuming flat fee) for order j, j=1, 2 . . . , n    -   For a “sell” order j, the adjusted limit price w_(j) ^(a) equals        (1−w_(j)−f)    -   For a buy order j, the adjusted limit price w_(j) ^(a) equals        (w_(j)−f_(j))-   B: n×m matrix where B_(j,i) is a positive number if the jth order    requests a payout for the i^(th) state, and 0 otherwise. For digital    options, the positive number is one.    -   Each row j of B comprises a payout profile for order j.-   f_(j): transaction fee for order j, scalar (in basis points) added    to and subtracted from equilibrium price to obtain offer and bid    prices, respectively, and subtracted from and added to limit prices,    w_(j), to obtain adjusted limit price, w_(j) ^(a) for buy and sell    limit prices, respectively.

Unknown Variables

-   x: n×1 vector where x_(j) is the notional payout executed for order    j in equilibrium    -   Range: 0≦x_(j) ≦r_(j) for j=1, 2, . . . , n-   y: m×1 vector where y_(i) is the notional payout executed per    defined state I, i=1, 2, . . . , m    -   Definition: y≡B^(T)x-   T: positive scalar, not necessarily an integer.    -   T is the total invested premium (in value units) in the contract

$T = {{{\sum\limits_{i = 1}^{m}{y_{i}p_{i}}} + {\sum\limits_{i = 1}^{m}k_{i}}} = {{\sum\limits_{j = 1}^{n}{x_{j}\pi_{j}}} + {\sum\limits_{i = 1}^{m}k_{i}}}}$

-   T_(i): positive scalar, not necessarily an integer    -   T_(i) is the total invested premium (in value units) in state i-   p: m×1 vector where p_(i) is the price/probability for state I, i=1,    2, . . . , m

${{p_{i} \equiv {\frac{k_{i}}{T - y_{i}}\mspace{14mu} {for}\mspace{14mu} i}} = 1},2,\ldots \mspace{11mu},m$

-   π_(j): equilibrium price for order j-   π(x): B*p, an n×1 vector containing the equilibrium prices for each    order j.-   g: n×1 vector whose j element is g_(j) for j=1, 2, . . . , n    -   Definition: g≡B*p−w    -   Note B*p is the vector of market prices for order j denoted by        π_(j)    -   g is the difference between the market prices and the limit        prices

7.2 Elements of Example DBAR DOE Embodiment

In this embodiment (Section 7), traders submit orders during the DBARmarket or auction that include the following data: (1) an order payoutsize (denoted r_(j)), (2) a limit order price (denoted w_(j)), and (3)the defined states for which the desired digital option is in-the-money(denoted as the rows of the matrix B, as described in the previoussub-section). In this embodiment, all of the order requests are in theform of payouts to be received should the defined states over which therespective options are in-the-money occur. In Section 6, an embodimentwas described in which the order amounts are invested premium amounts,rather than the aforementioned payouts.

7.3 Mathematical Principles

In this embodiment of a DBAR DOE market or auction, traders are able tobuy and sell digital options and spreads. The fundamental contingentclaims of this market or auction are the smallest digital optionspreads, i.e., those that span a single strike price. For example, ademand-based market or auction, such as, for example, a DBAR auction ormarket, that offers digital call and put options with strike prices of30, 40, 50, 60, and 70 contains six fundamental states: the spread belowand including 30; the spread between 30 and 40 including 40; the spreadbetween 40 and 50 including 50; etc. As indicated in the previoussection, in this embodiment, p_(i) is the price of a single strikespread i and m is the number of fundamental single state spreads or“defined states.” For these single strike spreads, the followingassumptions are made:

DBAR DOE Assumptions for this Embodiment

$\begin{matrix}{7.3{.1}} & \; \\{{\sum\limits_{i = 1}^{m}p_{i}} = 1} & (1) \\{{{p_{i} > {0\mspace{14mu} {for}\mspace{14mu} i}} = 1},\ldots \mspace{11mu},m} & (2) \\{{{k_{i} > {0\mspace{14mu} {for}\mspace{14mu} i}} = 1},\ldots \mspace{11mu},m} & (3)\end{matrix}$

The first assumption, equation 7.3.1(1), is that the fundamental spreadprices sum to unity. This equation holds for this embodiment as well asfor other embodiments of the present invention. Technically, the sum ofthe fundamental spread prices should sum to the discount factor thatreflects the time value of money (i.e., the interest rate) prevailingfrom the time at which investors must pay for their digital options tothe time at which investors receive a payout from an in-the-money optionafter the occurrence of a defined state. For the purposes of thisdescription of this embodiment, the time value of money during thisperiod will be taken to be zero, i.e., it will be ignored so that thefundamental spread prices sum to unity. The second assumption, equation7.3.1(2), is that each price must be positive. Assumption 3, equation7.3.1(3), is that the DBAR DOE contract of the present embodiment isinitialized (see Section 6.7, above) with value units invested in eachstate in the amount of k_(i) (initial amount of value units invested forstate i).

Using the notation from Section 7.1, the Demand Reallocation Function(DRF) of this embodiment of an OPF is a canonical DRF (CDRF), settingthe total amount of investments that are allocated using multistateallocation techniques to the defined states equal to the total amount ofinvestment in the auction or market that is available (net of anytransaction fees) to allocate to the payouts upon determining thedefined state which has occurred. Alternatively, a non-canonical DRF maybe used in an OPF.

Under a CDRF, the total amount invested in each defined state is afunction of the price in that state, the total amount of notional payoutrequested for that state, and the initial amount of value units investedin the defined state, or:

T _(i) =p _(i) *y _(i) k _(i)  7.3.2

The ratio of the invested amounts in any two states is therefore equalto:

$\begin{matrix}{\frac{T_{i}}{T_{j}} = \frac{{p_{i}*y_{i}} + k_{i}}{{p_{j}*y_{i}} + k_{j}}} & {7.3{.3}}\end{matrix}$

As described previously, since each state price is equal to the totalinvestment in the state divided by the total investment over all of thestates (p_(i)=T_(i)/T and p_(j)=T_(j)/T), the ratio of the investmentamounts in each DBAR contingent claim defined state is equal to theratio of the prices or implied probabilities for the states, which,using the notation of Section 7.1, yields:

$\begin{matrix}{\frac{T_{i}}{T_{j}} = {\frac{{p_{i}*y_{i}} + k_{i}}{{p_{j}*y_{j}} + k_{j}} = \frac{p_{i}}{p_{j}}}} & {7.3{.4}}\end{matrix}$

Eliminating the denominators of the previous equation and summing over jyields:

$\begin{matrix}{{\sum\limits_{j = 1}^{m}{p_{j}\left( {{p_{i}*y_{i}} + k_{i}} \right)}} = {\sum\limits_{j = 1}^{m}{p_{i}*\left( {{p_{j}*y_{j}} + k_{j}} \right)}}} & {7.3{.5}}\end{matrix}$

Substitution for T into the above equation yields:

$\begin{matrix}{{\left( {{p_{i}*y_{i}} + k_{i}} \right)\left( {\sum\limits_{j = 1}^{m}p_{j}} \right)} = {p_{i}T}} & {7.3{.6}}\end{matrix}$

By the assumption that the state prices or probabilities sum to unityfrom Equation 7.3.1, this yields the following equation:

$\begin{matrix}{p_{i} = \frac{k_{i}}{T - y_{i}}} & {7.3{.7}}\end{matrix}$

This equation yields the state price or probability of a defined statein terms of: (1) the amount of value units invested in each state toinitialize the DBAR auction or market (10; (2) the total amount ofpremium invested in the DBAR auction or market (T); and (3) the totalamount of payouts to be executed for all of the traders' orders forstate i (y_(i)). Thus, in this embodiment, Equation 7.3.7 follows fromthe assumptions stated above, as indicated in the equations in 7.3.1,and the requirement the DRF imposes that the ratio of the state pricesfor any two defined states in a DBAR auction or market be equal to theratio of the amount of invested value units in the defined states, asindicated in Equation 7.3.4.

7.4 Equilibrium Algorithm

From equation 7.3.7 and the assumption that the probabilities of thedefined states sum to one (again ignoring any interest rateconsiderations), the following m+1 equations may be solved to obtain theunique set of defined state probabilities (p's) and the total premiuminvestment for the group of defined states or contingent claims:

$\begin{matrix}{{p_{i} = \frac{k_{i}}{T - y_{i}}},{i = 1},2,\ldots \mspace{14mu},m} & (a) & {7.4{.1}} \\{{\sum\limits_{i = 1}^{m}p_{i}} = {{\sum\limits_{i = 1}^{m}\frac{k_{i}}{T - y_{i}}} = 1}} & (b) & \;\end{matrix}$

Equation 7.4.1 contains m+1 unknowns and m+1 equations. The unknowns arethe p_(i), i=1, 2, . . . , m, and T, the total investment for all of thedefined states. In accordance with the embodiment, the method ofsolution of the m+1 equations is to first solve Equation 7.4.1 (b). Thisequation is a polynomial in T. By the assumption that all of theprobabilities of the defined states must be positive, as stated inEquation 7.3.1, and that the probabilities also sum to one, as alsostated in Equation 7.3.1, the defined state probabilities are between 0and 1 or:

$\begin{matrix}{{{0 < p_{i} < 1},{{which}\mspace{14mu} {implies}}}{{0 < \frac{k_{i}}{T - y_{i}} < 1},{{{for}\mspace{14mu} i} = 1},2,{\ldots \mspace{14mu} m},{{which}\mspace{14mu} {implies}}}{{T > {y_{i} + k_{i}}},{{{for}\mspace{14mu} i} = 1},2,{\ldots \mspace{14mu} m},{{which}\mspace{14mu} {implies}}}{{T > {\max \left( {y_{i} + k_{i}} \right)}},{{{for}\mspace{14mu} i} = 1},2,{\ldots \mspace{20mu} m}}} & {7.4{.2}}\end{matrix}$

So the lower bound for T is equal to:

T _(lower)=max(y _(i) +k _(i))

By Equation 7.3.2:

$\begin{matrix}{T = {{\sum\limits_{i = 1}^{m}T_{i}} = {{\sum\limits_{i = 1}^{m}k_{i}} + {\sum\limits_{i = 1}^{m}{p_{i}y_{i}}}}}} & {7.4{.3}}\end{matrix}$

Letting y_((m)) be the maximum value of the y's,

$\begin{matrix}\begin{matrix}{T = {{{\sum\limits_{i = 1}^{m}k_{i}} + {\sum\limits_{i = 1}^{m}{p_{i}y_{i}}}} \leq {{\sum\limits_{i = 1}^{m}k_{i}} + {\sum\limits_{i = 1}^{m}{p_{i}y_{(m)}}}}}} \\{= {{\sum\limits_{i = 1}^{m}k_{i}} + {y_{(m)}{\sum\limits_{i = 1}^{m}p_{i}}}}} \\{= {{\sum\limits_{i = 1}^{m}k_{i}} + y_{(m)}}}\end{matrix} & {7.4{.4}}\end{matrix}$

Thus, the upper bound for T is equal to:

$\begin{matrix}{T_{upper} = {{{\sum\limits_{i = 1}^{m}k_{i}} + y_{(m)}} = {{\max \left( y_{i} \right)} + {\sum\limits_{i = 1}^{m}k_{i}}}}} & {7.4{.5}}\end{matrix}$

The solution for the total investment in the defined states thereforelies in the following interval

$\begin{matrix}{{{T_{lower} < T \leq T_{upper}},{or}}{{\max \left( {y_{i} + k_{i}} \right)} < T \leq {{\max \left( y_{i} \right)} + {\sum\limits_{i = 1}^{m}k_{i}}}}} & {7.4{.6}}\end{matrix}$

In this embodiment, T is determined uniquely from the equilibriumexecution order amounts, denoted by the vector x. Recall that in thisembodiment, y≡B^(T)x. As shown above,

Tε(T _(lower) ,T _(upper)]

Let the function f be

${f(T)} = {{{\sum\limits_{i = 1}^{m}\left( \frac{k_{i}}{T - y_{i}} \right)} - 1} = {0 = {{\sum\limits_{i = 1}^{m}p_{i}} - 1}}}$

Further,

f(T _(lower))>0

f(T _(upper))<0

Now, over the range T ε(T_(lower), T_(upper)], f(T) is differentiableand strictly monotonically decreasing. Thus, there is a unique T in therange such that

f(T)=0

Thus, T is uniquely determined by the x_(j)'s (the equilibrium executednotional payout amounts for each order j).

The solution for Equation 7.4.1(b) can therefore be obtained usingstandard root-finding techniques, such as the Newton-Raphson technique,over the interval for T stated in Equation 7.4.6. Recall that thefunction f(T) is defined as

${f(T)} = {{\sum\limits_{i = 1}^{m}\left( \frac{k_{i}}{T - y_{i}} \right)} - 1}$

The first derivative of this function is therefore:

${f^{\prime}(T)} = {\frac{f}{T} = {- {\sum\limits_{i = 1}^{m}\frac{k_{i}}{\left( {T - y_{i}} \right)^{2}}}}}$

Thus for T, take for an initial guess

T ⁰=Max(y ₁ +k ₁ ,y ₂ +k ₂ , . . . ,y _(m) +k _(m))

For the p+1^(st) guess use

$T^{p + 1} = {T^{p} - \frac{f\left( T^{p} \right)}{f^{\prime}\left( T^{p} \right)}}$

and calculate iteratively until a desired level of convergence to theroot of f(T), is obtained.

Once the solution for Equation 7.4.1(b) is obtained, the value of T canbe substituted into each of the m equations in 7.4.1(a) to solve for thep_(i). When the T and the p_(i) are known, all prices for DBAR digitaloptions and spreads may be readily calculated, as indicated by thenotation in 7.1.

Note that, in the alternative embodiment with no limit orders (brieflydiscussed at the beginning of this section 7), there are no constraintsset by limit prices, and the above equilibrium algorithm is easilycalculated because x_(j), the executed notional payout amounts for eachorder j, is equal to r_(j) (a known quantity), the requested notionalpayout for order j.

Regardless of the presence of limit orders, an equivalent set ofmathematics for this embodiment of a DBAR DOE is developed using matrixnotation. The matrix equivalent of Equation 7.3.2 may be written asfollows:

H*p=T*p  7.4.7

where T and p are the total premium and state probability vector,respectively, as described in Section 7.1. The matrix H, which has mrows and m columns where m is the number of defined states in the DBARmarket or auction, is defined as follows:

$\begin{matrix}{H = \begin{bmatrix}{y_{1} + k_{1}} & k_{1} & k_{1} & \ldots & k_{1} \\k_{2} & {y_{2} + k_{2}} & k_{2} & \ldots & k_{2} \\\vdots & \vdots & \vdots & \ldots & \vdots \\\; & \; & \; & \; & \; \\k_{m} & k_{m} & k_{m} & \ldots & {y_{m} + k_{m}}\end{bmatrix}} & {7.4{.8}}\end{matrix}$

H is a matrix with m rows and m columns. Each diagonal entry of H isequal to y_(i)+k_(i) (the sum of the notional payout requested by allthe traders for state i and the initial amount of value units investedfor state i). The other entries for each row are equal to k_(i) (theinitial amount of value units invested for state i). Equation 7.4.7 isan eigenvalue problem, where:

H=Y+K*V

Y=an m×m diagonal matrix of the aggregate notional amounts to beexecuted, Y_(i,i)=y_(i)

K=an m×m diagonal matrix of the arbitrary amounts of opening orders,K_(i,i)=k_(i)

V=an m×m matrix of ones, V_(i,j)=1

T=max (λ_(i)(H)), i.e., the maximum eigenvalue of the matrix H; and

p=|v(H,T)|, i.e., the normalized eigenvector associated with theeigenvalue T.

Thus, Equation 7.4.7 is, in this embodiment, a method of mathematicallydescribing the equilibrium of a DBAR digital options market or auctionthat is unique given the aggregate notional amounts to be executed (Y)and arbitrary amounts of opening orders (K). The equilibrium is uniquesince a unique total premium investment, T, is associated with a uniquevector of equilibrium prices, p, by the solution of the eigensystem ofEquation 7.4.7.

7.5 Sell Orders

In this embodiment, “sell” orders in a DBAR digital options market orauction are processed as complementary buy orders with limit pricesequal to one minus the limit price of the “sell” order. For example, forthe MSFT Digital Options auction of Section 6, a sell order for the 50calls with a limit price of 0.44 would be processed as a complementarybuy order for the 50 puts (which are complementary to the 50 calls inthe sense that the defined states which are spanned by the 50 puts arethose which are not spanned by the 50 calls) with limit price equal to0.56 (i.e., 1-0.44). In this manner, buy and sell orders, in thisembodiment of this Section 7, may both be entered in terms of notionalpayouts. Selling a DBAR digital call, put or spread for a given limitprice of an order j (w_(j)) is equivalent to buying the complementarydigital call, put, or spread at the complementary limit price of order j(1-w_(j)).

7.6 Arbitrary Payout Options

In this embodiment, a trader may desire an option that has a payoutshould the option expire in the money that varies depending upon whichdefined in-the-money state occurs. For example, a trader may desiretwice the payout if the state [40,50) occurs than if the state [30,40)occurs. Similarly, a trader may desire that an option have a payout thatis linearly increasing over the defined range of in-the-money states(“strips” as defined in Section 6 above) in order to approximate thetypes of options available in non-DBAR, traditional markets. Optionswith arbitrary payout profiles can readily be accommodated with the DBARmethods of the present invention. In particular, the B matrix, asdescribed in Section 7.2 above, can readily represent such options inthis embodiment. For example, consider a DBAR contract with 5 definedstates. If a trader desires an option that has the payout profile(0,0,1,2,3), i.e., an option that is in-the-money only if the last 3states occur, and for which the fourth state has a payout twice thethird, and the fifth state a payout three times the third, then the rowof the B matrix corresponding to this order is equal to (0,0,1,2,3). Bycontrast, a digital option for which the same three states arein-the-money would have a corresponding entry in the B matrix of(0,0,1,1,1). Additionally, for digital options all prices, bothequilibrium market prices and limit prices, are bound between 0 and 1.This is because all options are equally weighted linear combinations ofthe defined state probabilities. If, however, options with arbitrarypayout distributions are processed, then the linear combinations (asbased upon the rows of the B matrix) will not be weighted equally andprices need not be bounded between 0 and 1. For ease of exposition, thebulk of the disclosure in this Section 7 has assumed that digitaloptions (i.e., equally weighted payouts) are the only options underconsideration.

7.7 Limit Order Book Optimization

In this embodiment of a DBAR digital options exchange or market orauction as described in this Section 7, traders may enter orders fordigital calls, puts, and spreads by placing conditional investment orlimit orders. As indicated previously in Section 6.8, a limit order isan order to buy or sell a digital call, put or spread that contains aprice (the “limit price”) worse than which the trader desires not tohave his order executed. For example, for a buy order of a digital call,put, or spread, a limit order will contain a limit price which indicatesthat execution should occur only if the final equilibrium price of thedigital call, put or spread is at or below the limit price for theorder. Likewise, a limit sell order for a digital option will contain alimit price which indicates that the order is to be executed if thefinal equilibrium price is at or higher than the limit sell price. Allorders are processed as buy orders and are subject to execution wheneverthe order's limit price is greater than or equal to the then prevailingequilibrium price, because sell orders may be represented as buy orders,as described in the previous section.

In this embodiment, accepting limit orders for a DBAR digital optionsexchange uses the solution of a nonlinear optimization problem (oneexample of an OPF). The problem seeks to maximize the sum total ofnotional payouts of orders that can be executed in equilibrium subjectto each order's limit price and the DBAR digital options equilibriumEquation 7.4.7. Mathematically, the nonlinear optimization thatrepresents the DBAR digital options market or auction limit order bookmay be expressed as follows:

$\begin{matrix}{{x^{*} = {\underset{x}{argmax}{\sum\limits_{j = 1}^{n}x_{j}}}}{{subject}\mspace{14mu} {to}}} & {7.7{.1}} \\{{g_{j}(x)} = {{x_{j}\left( {{\pi_{j}(x)} - w_{j}^{a}} \right)} \leq 0}} & (1) \\{0 \leq x_{j} \leq r_{j}} & (2) \\{{Hp} = {Tp}} & (3)\end{matrix}$

The objective function of the optimization problem in 7.7.1 is the sumof the payout amounts for all of the limit orders that may be executedin equilibrium. The first constraint, 7.7.1(1), requires that the limitprice be greater than or equal to the equilibrium price for any payoutto be executed in equilibrium (recalling that all orders, including“sell” orders, may be processed as buy orders). The second constraint,7.7.1(2), requires that the execution payout for the order be positiveand less than or equal to the requested payout of the order. The thirdconstraint, 7.7.1(3) is the DBAR digital option equilibrium equation asdescribed in Equation 7.4.7.

7.8 Transaction Fees

In this embodiment, before solving the nonlinear optimization problem,the limit order prices for “sell” orders provided by the trader areconverted into buy orders (as discussed above) and both buy and “sell”limit order prices are adjusted with the exchange fee or transactionfee, f_(j). The transaction fee can be set for zero, or it can beexpressed as a flat fee as set forth in this embodiment which is addedto the limit order price received for “sell” orders, and subtracted fromthe limit order price paid for buy orders to arrive at an adjusted limitorder price w_(j) ^(a) for order j, as follows:

For a “sell” order j,w _(j) ^(a)=1−w _(j) −f _(i)  7.8.1

For a buy order j,w _(j) ^(f=w) _(j) −f _(j)  7.8.2

Alternatively, if the transaction fee f_(j) is variable, and expressedas a percentage of the limit order price, w_(j), then the limit orderprice may be adjusted as follows:

For a “sell” order j,

w _(j) ^(a)=(1−w _(j))*(1−f _(j))  7.8.3

For a buy order j,

w _(j) ^(a) =w _(j)*(1−f _(j))  7.8.4

The transaction fee f_(j) can also depend on the time of trade, toprovide incentives for traders to trade early or to trade certainstrikes, or otherwise reflect liquidity conditions in the contract.Regardless of the type of transaction fee f_(j), the limit order pricesw_(j) should be adjusted to w_(j) ^(a) before beginning solution of thenonlinear optimization program. Adjusting the limit order price adjuststhe location of the outer boundary for optimization set by the limitingequation 7.7.1(1). After the optimization solution has been reached, theequilibrium prices for each executed order j, π_(j)(x) can be adjustedby adding the transaction fee to the equilibrium price to produce themarket offer price, and by subtracting the transaction fee from theequilibrium price to produce the market bid price.

7.9 An Embodiment of the Algorithm to Solve the Limit Order BookOptimization

In this embodiment, the solution of Equation 7.7.1 can be achieved witha stepping iterative algorithm, as described in the following steps:

-   -   (1) Place Opening Orders: For each state, premium equal to        k_(i), for i=1, 2, . . . , m, is invested. These investments are        called the “opening orders.” The size of such investments, in        this embodiment, are generally small relative to the subsequent        orders.    -   (2) Convert all “sale” orders to complementary buy orders. As        indicated previously in Section 6.8, this is achieved by (i)        identifying the range of defined states i complementary to the        states being “sold”; and (ii) adjusting the limit “price”        (w_(j)) to one minus the original limit “price” (1−w_(j)). Note        that by contrast to the method disclosed in Section 6.8, there        is no need to convert the amount being sold into an equivalent        amount being bought. In this embodiment in this section, both        buy and “sell” orders are expressed in terms of payout (or        notional payout) terms.    -   (3) For all limit orders, adjust the limit “prices” (w_(j),        1−w_(j)) with transaction fee, by subtracting the transaction        fee f_(j): For a “sell” order j, the adjusted limit price w_(j)        ^(a) therefore equals (1−w_(j)+f_(j)), while for a buy order j,        the adjusted limit price w_(j) ^(a) equals (w_(j)−f_(j)).    -   (4) As indicated above in Section 6.8, group the limit orders by        placing all of the limit orders that span or comprise the same        range of defined states into the same group. Sort each group        from the best (highest “price” buy) to the worst (lowest “price”        buy).    -   (5) Establish an initial iteration step size, α_(j)(1). In this        embodiment the initial iteration step size α_(j)(1) may be        chosen to bear some reasonable relationship to the expected        order sizes to be encountered in the DBAR digital options market        or auction. In most applications, an initial iteration step size        α_(j)(1) equal to 100 is adequate. The current step size        α_(j)(κ) will initially equal the initial iteration step size        (α_(j)(κ)=α_(j)(1) for first iteration) until and unless the        current step size is adjusted to a different step size.    -   (6) Calculate the equilibrium to obtain the total investment        amount T and the state probabilities, p, using equation 7.4.7.        Although the eigenvalues can be computed directly, this        embodiment finds T by Newton-Raphson solution of Equation        7.4.1(b). The solution to T and equation 7.4.1(a) is used to        find the p's.    -   (7) Compute the equilibrium order prices π(x) using the p's        obtained in step (5). The equilibrium order prices π(x) are        equal to B*p.    -   (8) Increment the orders (x_(j)) that have adjusted limit prices        (w_(j) ^(a)) greater than or equal to the current equilibrium        price for that order π_(j)(x) (obtained in step (6)) by the        current step size α_(j)(κ), but not to exceed the requested        notional payout of the order, r_(j). Decrement the orders        (x_(j)) that have a positive executed order amount (x_(j)>0) and        have limit prices less than the current equilibrium market price        π_(j)(x) by the current step size α_(j)(κ), but not to an amount        less than zero.

(9) Repeat steps (5) to (7) in subsequent iterations until the valuesobtained for the executed order amounts (x_(j)'s) achieve a desiredconvergence, as measured by certain convergence criteria (set forth inStep (8)a), periodically adjusting the current step size α_(j)(κ) and/orthe iteration process after the initial iteration to further progressthe stepping iterative process towards the desired convergence. Theadjustments are set forth in steps (8)b to (8)d.

-   -   (8)a In this embodiment, the stepping iterative algorithm is        considered converged based upon a number of convergence        criteria. One such criterion is a convergence of the state        probabilities (“prices”) of the individual defined states. A        sampling window can be chosen, similar to the method by which        the rate of progress statistic is measured (described below), in        order to measure whether the state probabilities are fluctuating        or are merely undergoing slight oscillations (say at the level        of 10⁻⁵) that would indicate a tolerable level of convergence.        Another convergence criterion, in this embodiment, would be to        apply a similar rate of progress statistic to the order steps        themselves. Specifically, the iterative stepping algorithm may        be considered converged when all of the rate of progress        statistics in Equation 7.9.1(c) below are tolerably close to        zero. As another convergence criterion, in this embodiment, the        iterative stepping algorithm will be considered converged when,        in, possible combination with other convergence criteria, the        amount of payouts to be paid should any given defined state        occur does not exceed the total amount of investment in the        defined states, T, by a tolerably small amount, such as 10⁻⁵*T.    -   (8)b In this embodiment, the step size may be increased and        decreased dynamically based upon the experienced progress of the        iterative scheme. If, for example, the iterative increments and        decrements are making steady linear progress, then it may be        advantageous to increase the step size. Conversely, if the        iterative increments and decrements (“stepping”) is making less        than linear progress or, in the extreme case, is making little        or no progress, then it is advantageous to reduce the size of        the iterative step.        -   In this embodiment, the step size may be accelerated and            decelerated using the following:

$\begin{matrix}{\omega = {\mu*\theta}} & (a) & {7.9{.1}} \\{{{{mod}\left( \frac{\kappa}{\omega} \right)} = 0},{\kappa > \omega}} & (b) & \; \\{{\gamma_{j}(\kappa)} = \frac{{{x_{j}(\kappa)} - {x_{j}\left( {\kappa - \omega} \right)}}}{\sum\limits_{i = 1}^{\omega}{{{x_{j}(i)} - {x_{j}\left( {i - 1} \right)}}}}} & (c) & \; \\{\alpha_{j} = \left\{ \begin{matrix}{{\theta^{(\frac{{{\gamma_{j}{(\kappa)}}*\theta} - 1}{\theta - 1})}*{\alpha_{j}\left( {\kappa - 1} \right)}},{{\gamma_{j}(\kappa)} > \frac{1}{\theta}}} \\{{\theta^{{{\gamma_{j}{(\kappa)}}*\theta} - 1}*{\alpha_{j}\left( {\kappa - 1} \right)}},{{\gamma_{j}(\kappa)} \leq \frac{1}{\theta}}}\end{matrix} \right.} & (d) & \;\end{matrix}$

-   -   where Equation 7.9.1(a) contains the parameters of the        acceleration/deceleration rules. These parameters have the        following interpretation:    -   θ: a parameter that controls the rate of step size acceleration        and deceleration. Typically, the values for this parameter will        range between 2 and 4, indicating that a maximum range of        acceleration from 100-300%.    -   μ: a multiplier parameter, which, when used to multiply the        parameter θ, yields a number of iterations over which the step        size remains unchanged. Typically, the range of values for this        parameter are 3 to 10.    -   ω: the window length parameter, which is the product of θ and μ        over which the step size remains unchanged. The window parameter        is a number of iterations over which the orders are stepped with        a fixed step size. After these number of iterations, the        progress is assessed, and the step size for each order may be        accelerated or decelerated. Based upon the above described        ranges for θ and μ, the range of values for ω is between 6 and        40, i.e., every 6 to 40 iterations the step size is evaluated        for possible acceleration or deceleration.    -   κ: the variable denoting the current iteration of the step        algorithm where κ is an integer multiple of the window length,        ω.    -   γ_(j)(κ): a calculated statistic, calculated at every κ^(th)        iteration for each order j. The statistic is a ratio of two        quantities. The numerator is the absolute value of the        difference between the quantity of order j filled at the        iteration corresponding to the beginning of the window and at        the iteration at the end of window. It represents, for each        order j, the total amount of progress made, in terms of the        execution of order j by either incrementing or decrementing the        executed quantity of order j, from the start of the window to        the end of the window iteration. The denominator is the sum of        the absolute changes of the order execution for each iteration        of the window. Thus, if an order has made no progress, the        γ_(j)(κ) statistic will be zero. If each step has resulted in        progress in the same direction the γ_(j)(κ) statistic will equal        one. Thus, in this embodiment, the γ_(j)(κ) statistic represents        the amount of progress that has been made over the previous        iteration window, with zero corresponding to no progress for        order j and one corresponding to linear progress for order j.    -   α_(j)(κ): this parameter is the current step size for order j at        iteration count κ. At every κ^(th) iteration, it is updated        using the equation 7.9.1(d). If the γ_(j)(κ) statistic reflects        sufficient progress over the previous window by exceeding the        quantity 1/θ, then 7.9.1(d) provides for an increase in the step        size, which is accomplished through a multiplication of the        current step size by a number exceeding one as governed by the        formula in 7.9.1(d). Similarly, if the γ_(j)(κ) statistic        reflects insufficient progress by being equal or less than 1/θ,        the step size parameter will remain the same or will be reduced        according to the formula in 7.9.1(d).    -   These parameters are selected, in this embodiment, based upon,        in part, the overall performance of the rules with respect to        test data. Typically, θ=2-4, μ=3-10 and therefore ω=6-40.        Different parameters may be selected depending upon the overall        performance of the rules. Equation 7.9.1(b) states that the        acceleration or deceleration of an iterative step for each        order's executed amount is to be performed only on the ω-th        iteration, i.e., ω is a sampling window of a number of        iterations (say 6-40) over which the iterative stepping        procedure is evaluated to determine its rate of progress.        Equation 7.9.1(c) is the rate of progress statistic that is        calculated over the length of each sampling window. The        statistic is calculated for each order j on every ω-th iteration        and measures the rate of progress over the previous ω iterations        of stepping. For each order, the numerator is the absolute value        of how much each order j has been stepped over the sampling        window. The larger the numerator, the larger the amount of total        progress that has been made over the window. The denominator is        the sum of the absolute values of the progress made over each        individual step within the window, summed over the number of        steps, ω, in the window. The denominator will be the same value,        for example, whether 10 positive steps of 100 have been made or        whether 5 positive steps of 100 and 5 negative steps of 100 have        been made for a given order. The ratio of the numerator and        denominator of Equation 7.9.1(c) is therefore a statistic that        resides on the interval between 0 and 1, inclusive. If, for        example, an order j has not made any progress over the window        period, then the numerator is zero and the statistic is zero.        If, however, an order j has made maximum progress over the        window period, the rate of progress statistic will be equal        to 1. Equation 7.9.1(d) describes the rule based upon the rate        of progress statistic. For each order j at iteration κ (where κ        is a multiple of the window length), if the rate of progress        statistic exceeds 1/θ, then the step size is accelerated. A        higher choice of the parameter θ will result in more frequent        and larger accelerations. If the rate of progress statistic is        less than or equal to 1/θ, then the step size is either kept the        same or decelerated. It may be possible to employ similar and        related acceleration and deceleration rules, which may have a        somewhat different mathematical parameterization as that        described above, to the iterative stepping of the order amount        executions.    -   (8)c In this embodiment, a linear program may be used, in        conjunction with the iterative stepping algorithm described        above, to further accelerate the rate of progress. The linear        program would be employed primarily at the point when a        tolerable level of convergence in the defined state        probabilities has been achieved. When the defined state        probabilities have reached a tolerable level of convergence, the        nonlinear program of Equation 7.7.1 is transformed, with prices        held constant, into a linear program. The linear program may be        solved using widely available techniques and software code. The        linear program may be solved using a variety of numerical        tolerances on the set of linear constraints. The linear program        will yield a result that is either feasible or infeasible. The        result contains the maximum sum of the executed order amounts        (sum of the x_(j)), subject to the price, bounds, and        equilibrium constraints of Equation 7.7.1, but with the prices        (the vector p) held constant. In frequent cases, the linear        program will result in executed order amounts that are larger        than those in possession at the current iteration of the        stepping procedure. After the linear program is solved, the        iterative stepping procedure is resumed with the executed order        amounts from the linear program. The linear program is an        optimization program of Equation 7.7.1 but with the vector p        from the current iteration κ held constant. With prices        constant, constraints (1) and (3) of nonlinear optimization        problem 7.7.1 become linear and therefore Equation 7.7.1 is        transformed from a nonlinear optimization program to a linear        program.    -   (8)d Once a tolerable level of convergence has been achieved for        the notional payout executed for each order, x_(j), the entire        stepping iterative algorithm to solve Equation 7.7.1 may then be        repeated with a substantially smaller step size, e.g., a step        size, α_(j)(κ), equal to 1 until a higher level of convergence        has been achieved.

7.10 Limit Order Book Display

In this embodiment of a DBAR digital options market or auction, it maybe desirable to inform market or auction participants of the amount ofpayout that could be executed at any given limit price for any givenDBAR digital call, put, or spread, as described previously in Section6.9. The information may be displayed in such a manner so as to informtraders and other market participants the amount of an order that may bebought and “sold” above and below the current market price,respectively, for any digital call, put, or spread option. In thisembodiment, such a display of information of the limit order bookappears in a manner similar to the data displayed in the followingtable.

TABLE 7.10.1 Current Pricing Strike Spread To Bid Offer Payout Volume<50 0.2900 0.3020 3.3780 110,000,000 <50 PUT Offer Offer Side Volume0.35 140,002,581 0.32 131,186,810 0.31 130,000,410 MARKET PRICE 0.29000.3020 MARKET PRICE 120,009,731 0.28 120,014,128 0.27 120,058,530 0.24Bid Side Volume BidIn Table 7.10.1, the amount of payout that a trader could execute werehe willing to place an order at varying limit prices above the market(for buy orders) and below the market (for “sell” orders) is displayed.As displayed in the table, the data pertains to a put option, say forMSFT stock as in Section 6, at a strike price of 50. The current priceis 0.2900/0.3020 indicating that the last “sale” order could have beenprocessed at 0.2900 (the current bid price) and that the last buy ordercould have been processed at 0.3020 (the current offer price). Thecurrent amount of executed notional volume for the 50 put is equal to110,000,000. The data indicate that a trader willing to place a buyorder with limit price equal to 0.31 would be able to executeapproximately 130,000,000 notional payout. Similarly, a trader willingto place a “sell” order with limit price equal to 0.28 would be able toachieve indicative execution of approximately 120,000,000 in notional.

7.11 Unique Price Equilibrium Proof

The following is a proof that a solution to Equation 7.7.1 results in aunique price equilibrium. The first-order optimality conditions forEquation 5 yield the following complementary conditions:

(1)g _(j)(x)<0→x _(j) =r _(j)

(2)g _(j)(x)>0→x _(j)=0

(3)g _(j)(x)=0→0≦x _(j) ≦r _(j)

The first condition is that if an order's limit price is higher than themarket price (g_(j)(x)<0), then that order is fully filled (i.e., filledin the amount of the order request, r_(j)). The second condition is thatan order not be filled if the order's limit price is less than themarket equilibrium price (i.e., g_(j)(x)>0). Condition 3 allows fororders to be filled in all or part in the case where the order's limitprice exactly equals the market equilibrium price.

To prove the existence and convergence to a unique price equilibrium,consider the following iterative mapping:

F(x)=x−β*g(x)  7.11.2A

Equation 7.11.2A can be proved to be contraction mapping which for astep size independent of x will globally converge to a uniqueequilibrium, i.e., it can be proven that Equation 2A has a unique fixedpoint of the form

F(x*)=x*  7.11.3A

To first show that F(x) is a contraction mapping, matrix differentiationof Equation 2A yields:

$\begin{matrix}{{\frac{{F(x)}}{x} = {I - {\beta^{*}{D(x)}}}}{where}{{D(x)} = {B*A*Z^{- 1}*B^{T}}}{A_{i,j} = \left\{ {{\begin{matrix}{{p_{i}*\left( {1 - p_{i}} \right)},} & {i = j} \\{{{- p_{i}}*p_{j}},} & {i \neq j}\end{matrix}Z_{i,j}} = \left\{ \begin{matrix}{{T - y_{i} + {p_{i}*y_{i}}},} & {i = j} \\{{p_{j}*y_{i}},} & {i \neq j}\end{matrix} \right.} \right.}} & {7.11{.4}A}\end{matrix}$

The matrix D(x) of Equation 4A is the matrix of order price firstderivatives (i.e., the order price Jacobian). Equation 7.11.2A can beshown to be a contraction if the following condition holds:

$\begin{matrix}{{\frac{{F(x)}}{x}} < 1} & {7.11{.5}A}\end{matrix}$

which is the case if the following condition holds:

β*ρ(D)<1,

where

ρ(D)=max(λ_(i),(D)), i.e., the spectral radius of D  7.11.6A

By the Gerschgorin's Circle Theorem the eigenvalues of A are boundedbetween 0 and 1. The matrix Z⁻¹ is a diagonally dominant matrix, allrows of which sum to 1/T. Because of the diagonal dominance, the othereigenvalues of Z⁻¹ are clustered around the diagonal elements of thematrix, and are approximately equal to p_(i)/k_(i). The largesteigenvalue of Z⁻¹ is therefore bounded above by 1/k_(i). The spectralradius of D is therefore bounded between 0 and linear combinations of1/k_(i) as follows:

$\begin{matrix}{{{\rho (D)} \leq L}{L = \frac{1}{\sum\limits_{i = 1}^{m}\frac{1}{k_{i}}}}} & {7.11{.7}A}\end{matrix}$

where the quantity L, a function of the opening order amounts, can beinterpreted as the “liquidity capacitance” of the demand-based tradingequilibrium (mathematically L is quite similar to the total capacitanceof capacitors in series). The function F(x) of Equation 2A is thereforea contraction if

β<L  7.11.8A

Equation 7.11.8A states that a contraction to the unique priceequilibrium can be guaranteed for contraction step sizes no larger thanL, which is an increasing function of the opening orders in thedemand-based market or auction.

The fixed point iteration of Equation 2A converges to x*. Sincey*=B^(T)x*, y* can be used in Equation 7.4.7 to compute the fundamentalstate prices p* and the total quantity of premium invested T*. If thereare linear dependencies in the B matrix, it may be possible to preservep* through a different allocation of the x's corresponding to thelinearly dependent rows of B. For example, consider two orders, x₁ andx₂, which span the same states and have the same limit order price.Assume that r₁=100 and r₂=100 and that x₁*=x₂*=50 from the fixed pointiteration. Then clearly, x₁=100 and x₂=0 may be set without disturbingp*. For example, different order priority rules may give executionprecedence to the earlier submitted identical order. In any event, thefixed point iteration results in a unique price equilibrium, that is,unique in p.

8. Network Implementation

A network implementation of the embodiment described in Section 7 is ameans to run a complete, market-neutral, self-hedging open book of limitorders for digital options. The network implementation is formed from acombination of demand-based trading core algorithms with an electronicinterface and a demand-based limit order book. This embodiment enablesthe exchange or sponsor to create products, e.g., a series ofdemand-based auctions or markets specific to an underlying event, inresponse to customer demand by using the network implementation toconduct the digital options markets or auctions. These digital options,in turn, form the foundation for a variety of investment, riskmanagement and speculative strategies that can be used by marketparticipants. As shown in FIG. 22, whether accessed using secure,browser-based interfaces over web sites on the Internet or an extensionof a private network, the network implementation provides market makerswith all the functionality conduct a successful market or auctionincluding, for example:

-   (1) Order entry. Orders are taken by a market maker's sales force    and entered into the network implementation.-   (2) Limit order book. All limit orders are displayed.-   (3) Indicative pricing and volumes. While an auction or market is in    progress, prices and order volumes are displayed and updated in real    time.-   (4) Price publication. Prices may be published using the market    maker's intranet (for a private network implementation) or Internet    web site (for an Internet implementation) in addition to market data    services such as Reuters and Bloomberg.-   (5) Complete real-time distribution of market expectations. The    network implementation provides market participants with a display    of the complete distribution of expected returns at all times.-   (6) Final pricing and order amounts. At the conclusion of a market    or an auction, final prices and filled orders are displayed and    delivered to the market maker for entry or export to existing    clearing and settlement systems.-   (7) Auction or Market administration. The network implementation    provides all func-tions necessary to administer the market or    auction, including start and stop functions, and details and summary    of all orders by customer and salesperson.

A practical example of a demand-based market or auction conducted usingthe network implementation follows. The example assumes that aninvestment bank receives inquiries for derivatives whose payouts arebased upon a corporation's quarterly earnings release. At present, nounderlying tradable supply of quarterly corporate earnings exists andfew investment banks would choose to coordinate the “other side” of sucha transaction in a continuous market.

Establishing the Market or Auction: First, the sponsor of the market orauction establishes and communicates the details that define the marketor auction, including the following:

-   -   An underlying event, e.g., the scheduled release of an earnings        announcement    -   An auction period or trading period, e.g., the specified date        and time period for the market or auction    -   Digital options strike prices, e.g., the specified increments        for each strike        Accepting and Processing Customer Limit Orders: During the        auction or trading period, customers may place buy and sell        limit orders for any of the calls or puts, as defined in the        market or auction details establishing the market or auction.        Indicative and Final Clearing of the Limit Order Book: During        the auction or trading period, the network implementation        displays indicative clearing prices and quantities, i.e., those        that would exist if the order book were cleared at that moment.        The network implementation also displays the limit order book        for each option, enabling market participants to assess market        depth and conditions. Clearing prices and quantities are        determined by the available intersection of limit orders as        calculated according to embodiments of the present invention. At        the end of the auction or trading period, a final clearing of        the order book is performed and option prices and filled order        quantities are finalized. Market participants remit and accept        premium for filled orders. This completes a successful market or        auction of digital options on an event with no underlying        tradable supply.        Summary of Demand-Based Market or Auction Benefits: Demand-based        markets or auctions can operate efficiently without the        requirement of a discrete order match between and among buyers        and sellers of derivatives. The mechanics of demand-based        markets or auctions are transparent. Investment, risk management        and speculative demand exists for large classes of economic        events, risks and variables for which no associated tradable        supply exists. Demand-based markets and auctions meet these        demands.

9. Structured Instrument Trading

In another embodiment, clients can offer instruments suitable to broadclasses of investors. In particular, an opportunity exists forparticipation in demand-based markets or auctions by customers who wouldotherwise not participate because they typically avoid leverage andtrading in derivatives contracts. In this embodiment, these customersmay transact using existing financial instruments or other structuredproducts, for example, risk-linked notes and swaps, simultaneously withcustomers transacting using DBAR contingent claims, for example, digitaloptions, in the same demand-based market or auction.

In this embodiment, a set of one or more digital options are created toapproximate one or more parameters of the structured products, e.g., aspread to LIBOR (London Interbank Offered Rate) or a coupon on arisk-linked note or swap, a note notional (also referred to, forexample, as a face amount of the note or par or principal), and/or atrigger level for the note or swap to expire in-the-money. The set ofone or more digital options may be referred to, for example, as anapproximation set. The structured products become DBAR-enabled products,because, once their parameters are approximated, the customer is enabledto trade them alongside other DBAR contingent claims, for example,digital options.

The approximation, a type of mapping from parameters of structuredproducts to parameters of digital options, could be an automaticfunction built into a computer system accepting and processing orders inthe demand-based market or auction. The approximation or mapping permitsor enables non-leveraged customers to interface with the demand-basedmarket or auction, side by side with leverage-oriented customers whotrade digital options. DBAR-enabled notes and swaps, as well as otherDBAR-enabled products, provide non-leveraged customers the ability toenhance returns and achieve investment objectives in new ways, andincrease the overall liquidity and risk pricing efficiency of thedemand-based market or auction by increasing the variety and number ofparticipants in the market or auction.

9.1 Overview: Customer-Oriented DBAR-Enabled Products

Instruments can be offered to fit distinct investment styles, needs, andphilosophies of a variety of customers. In this embodiment, “clienteleeffects” refers to, for example, the factors that would motivatedifferent groups of customers to transact in one type of DBAR-enabledproduct over another. The following classes of customers may havevarying preferences, institutional constraints, and investment and riskmanagement philosophies relevant to the nature and degree ofparticipation in demand-based markets or auctions:

-   -   Hedge Funds

Proprietary Traders

Derivatives Dealers

Portfolio Managers

Insurers and Reinsurers

Pension Funds

Regulatory, accounting, internal institutional policies, and otherrelated constraints may affect the ability, willingness, and frequencyof participation in leveraged investments in general and derivativesproducts such as options, futures, and swaps in particular. Hedge fundsand proprietary traders, for instance, may actively trade digitaloptions, but may be unlikely to trade in certain structured noteproducts that have identical risks while requiring significant capital.On the other hand, “real money” accounts such as portfolio managers,insurers, and pension funds may actively trade instruments that bearsignificant event risk, but these real money customers may be unlikelyto trade DBAR digital options bearing identical event risks.

For example, according to the prospectus for their total return fund,one particular fixed income manager may invest in fixed incomesecurities for which the return of principal and payment of interest arecontingent upon the non-occurrence of a specific ‘trigger’ event, suchas a hurricane, earthquake, tornado, or other phenomenon (referred to,for example, as ‘event-linked bonds’). These instruments typically pay aspread to LIBOR should losses not exceed a stipulated level.

On the other hand, a fixed-income manager may not trade in an IndustryLoss Warranty market or auction with insurers (discussed above inSection 3), even though the risks transacted in this market or auction,effectively a market or auction for digital options on property risksposed by hurricanes, may be identical to the risks borne in theunderwritten Catastrophe-linked (CAT) securities. Similarly, thefixed-income manager and other fixed income managers may participatewidely in the corporate bond market, but may participate to a lesserextent in the default swap market (convertible into a demand-basedmarket or auction), even though a corporate bond bears similar risks asa default swap bundled with a traditional LIBOR-based note or swap.

The unifying theme to these clientele effects is that the structure andform in which products are offered can impact the degree of customerparticipation in demand-based markets or auctions, especially for realmoney customers which avoid leverage and trade few, if any, options butactively seek fixed-income-like instruments offering significant spreadsto LIBOR for bearing some event-related risk on an active and informedbasis.

This embodiment addresses these “clientele effects” in the risk-bearingmarkets by allowing demand-based markets or auctions to simultaneouslyoffer both digital options and DBAR-enabled products, such as, forexample, risk-linked FRNs (or floating rate notes) and swaps, todifferent customers within the same risk-pricing, allocation, andexecution mechanism. Thus, hedge funds, arbitrageurs, and derivativesdealers can transact in the demand-based market or auction in terms ofdigital options, while real money customers can transact in thedemand-based market or auction in terms of different sets ofinstruments: swaps and notes paying spreads to LIBOR. For both types ofcustomers, the payout is contingent upon an observed outcome of aneconomic event, for example, the level of the economic statistic at therelease date (or e.g., at the end of the observation period).

9.2 Overview: FRNs and swaps

For FRN and swap customers, according to this embodiment, a nexus ofcounterparties to contingent LIBOR-based cash flows based upon materialrisky events can be created in a demand-based market or auction.Schematically, the cash flows resemble a multiple counterparty versionof standard FRN or swap LIBOR-based cash flows. FIG. 23 illustrates thecash flows for each participant. The underlying properties of DBARmarkets or auctions will still apply (as described below), the offeringof this event-linked FRN is market-neutral and self-hedging. In thisembodiment, as with other embodiments of the present invention, ademand-based market or auction is created, ensuring that the receiversof positive spreads to LIBOR are being funded, and completely offset, bythose out-of-the-money participants who receive par.

In this example, with actual ECI at 0.9%, the participants, each withtrigger levels of 0.7%, 0.8%, or 0.9% are all in-the-money, and willearn LIBOR plus the corresponding spread for those triggers on. Thoseparticipants with trigger levels above 0.9% receive par.

9.3 Parameters: FRNs and Swaps vs. Digital Options

The following information provides an illustration of parameters relatedto a principal-protected Employment Cost Index(ECI)-linked FRN note andswap and ECI-linked digital options:

-   -   End of Trading Period: Oct. 23, 2001    -   End of Observation Period: Oct. 25, 2001    -   Coupon Reset Date: Oct. 25, 2001        -   (also referred to, for example, as the “FRN Fixing Date”)    -   Note Maturity: Jan. 25, 2002        -   (when par amount needs to be repaid)    -   Option payout date: Jan. 25, 2002        -   (when payout of digital option is paid, can be set to be the            same date as Note Maturity or a different date)    -   Trigger Index: Employment Cost Index (“ECI”)        -   (also known as the strike price for an equivalent DBAR            digital option)    -   Principal Protection: Par

TABLE 9.3 Indicative Trigger Levels and Indicative Pricing Spread toLIBOR* ECI Trigger (%) (bps) 0.7 50 0.8 90 0.9 180 1 350 1.1 800 1.21200 *For the purposes of the example, assume mid-market LIBOR execution

In this example, a customer (for example, an FRN holder or a noteholder) places an order for an FRN with $100,000,000 par (also referredto, for example, as the face value of the note or notional or principalof the note), selecting a trigger of 0.9% ECI and a minimum spread of180 bps to LIBOR (180 basis points or 1.80% in addition to LIBOR) duringa trading period. After the end of the trading period, Oct. 23, 2001, ifthe market or auction determines the coupon for the note (e.g., thespread to LIBOR) equal to 200 bps to LIBOR, and the customer's noteexpires in-the-money at the end of the observation period, Oct. 25,2001, then the customer will receive a return of 200 bps plus LIBOR onpar ($100,000,000) on the note maturity date, Jan. 25, 2002.

Alternatively, if the market or auction fixes the rate on the note orsets the spread to 180 bps to LIBOR, and the customer's noteexpires-in-the money at the end of the observation period, then thecustomer will receive a return of 180 bps plus LIBOR (the selectedminimum spread) on par on the note maturity date. If a 3-month LIBOR isequal to 3.5%, and the spread of 180 bps to LIBOR is also for a 3 monthperiod, and the note expires in-the-money, then the customer receives apayout $101,355,444.00 on Jan. 25, 2002, or:

$\begin{matrix}{{{in}\text{-}{the}\text{-}{money}\mspace{14mu} {payout}} = {{par} + {{par} \times \left( {{L\; I\; B\; O\; R} + {spread}} \right) \times \frac{daycount}{basis}}}} & {9.3A}\end{matrix}$

An “in-the-money note payout” may be a payout that the customer receivesif the FRN expires in-the-money. Analogously, an “out-of-the-money notepayout” may be a payout that the customer receives if the FRN expiresout-of-the-money. “Daycount” is the number of days between the end ofthe coupon reset date and the note maturity date (in this example, 92days). Basis is the number of days used to approximate a year, often setat 360 days in many financial calculations. The variable,“daycount/basis” is the fraction of a year between the observationperiod and the note maturity date, and is used to adjust the relevantannualized interest rates into effective interest rates for a fractionof a year.

If the note expires out-of-the-money, because the ECI is observed to be0.8%, for example, on Oct. 25, 2001 (the end of the observation period),then the customer receives an out-of-the-money payout of par on Jan. 25,2002, the note maturity date, or:

out-of-the-money note payout=par  9.3B

Alternatively, the FRN could be structured as a swap, in which case theexchange of par does not occur. If the swap is structured to adjust theinterest rates into effective interest rates for the actual amount oftime elapsed between the end of the observation period and the notematurity date, then the customer receives a swap payout of $1,355,444.If the ECI fixes below 0.9% (and the swap is structured to adjust theinterest rates), then the FRN holder loses or pays a swap loss of$894,444 or LIBOR times par (see equation 9.3D). The swap payout andswap loss can be formulated as follows:

$\begin{matrix}{{{swap}\mspace{14mu} {payout}} = {{par} \times \left( {{L\; I\; B\; O\; R} + {spread}} \right) \times \frac{daycount}{basis}}} & {9.3C} \\{{swaploss} = {{par} \times L\; I\; B\; O\; R \times \frac{daycount}{basis}}} & {9.3D}\end{matrix}$

As opposed to FRNs and swaps, digital options provide a notional or apayout at a digital payout date, occurring on or after the end of theobservation period (when the outcome of the underlying event has beenobserved). The digital payout date can be set at the same time as thenote maturity date or can occur at some other earlier time, as describedbelow. The digital option customer can specify a desired or requestedpayout, a selected outcome, and a limit on the investment amount forlimit orders (as opposed to market orders, in which the customer doesnot place a limit on the investment amount needed to achieve the desiredor requested payout).

9.4 Mechanics: DBAR-Enabling FRNs and Swaps

In this embodiment, as discussed above, both digital options andrisk-linked FRNs or swaps may be offered in the same demand-based marketor auction. Due to clientele effects, traditional derivatives customersmay follow the market or auction in digital option format, while thereal money customers may participate in the market or auction in an FRNformat. Digital options customers may submit orders, inputting optionnotional (as a desired payout), a strike price (as a selected outcome),and a digital option limit price (as a limit on the investment amount).FRN customers may submit orders, inputting a notional note size or par,a minimum spread to LIBOR, and a trigger level or levels, indicating thelevel (equivalently, a strike price) at or above which the FRN will earnthe market or auction-determined spread to LIBOR or the minimum spreadto LIBOR. An FRN may provide, for example, two trigger levels (or strikeprices) indicating that the FRN will earn a spread should the ECI Indexfall between them at the end of the observation period.

In this embodiment, the inputs for an FRN order (which are some of theparameters associated with an FRN) can be mapped or approximated, forexample, at a built-in interface in a computer system, into desiredpayouts, selected outcomes and limits on the investment amounts for oneor more digital options in an approximation set, so that the FRN ordercan be processed in the same demand-based market or auction along withdirect digital option orders. Specifically, each FRN order in terms of anote notional, a coupon or spread to LIBOR, and trigger level may beapproximated with a LIBOR-bearing note for the notional amount (or anote for notional amount earning an interest rate set at LIBOR), and anembedded approximation set of one or more digital options.

As a result of the mapping or approximation, all orders of contingentclaims (for example, digital option orders and FRN orders) are expressedin the same units or variables. Once all orders are expressed in thesame units or variables, an optimization system, such as that describedabove in Section 7, determines an optimal investment amount and executedpayout per order (if it expires in-the-money) and total amount investedin the demand-based market or auction. Then, at the interface, theparameters of the digital options in the approximation set correspondingto each FRN order are mapped back to parameters of the FRN order. Thecoupon for the FRN (if above the minimum spread to LIBOR specified bythe customer) is determined as a function of the digital options in theapproximation set which are filled and the equilibrium price of thefilled digital options in the approximation set, as determined by theentire demand-based market or auction. Thus, the FRN customer inputscertain FRN parameters, such as the minimum spread to LIBOR and thenotional amount for the note, and the market or auction generates otherFRN parameters for the customer, such as the coupon earned on thenotional of the note if the note expires-in-the money.

The methods described above and in section 9.5 below set forth anexample of the type of mapping that can be applied to the parameters ofa variety of other structured products, to enable the structuredproducts to be traded in a demand-based market or auction alongsideother DBAR contingent claims, including, for example, digital options,thereby increasing the degree and variety of participation, liquidityand pricing efficiency of any demand-based market or auction. Thestructured products include, for example, any existing or futurefinancial products or instruments whose parameters can be approximatedwith the parameters of one or more DBAR contingent claims, for example,digital options. The mapping in this embodiment can be used incombination with and/or applied to the other embodiments of the presentinvention.

9.5 Example Mapping FRNs into Digital Option Space

The following notation, figures and equations illustrate the mapping ofECI-linked FRNs into digital option space, or approximating theparameters of ECI-linked FRNs into parameters of an approximation set ofone or more digital options, and can be applied to illustrate themapping of ECI-linked swaps into digital option space.

9.5.1 Date and Timing Notation and Formulation

-   t_(S): the premium settlement date for the direct digital option    orders and the FRN orders, set at the same time or some time after    the TED (or the end of the trading or auction period).-   t_(E): the event outcome date or the end of the observation period    (e.g., the date of that the outcome of the event is observed).-   t_(O): the option payout date-   t_(R): the coupon reset date, or the date when interest (spread to    LIBOR, including, for example, spread plus LIBOR) begins to accrue    on the note notional.-   t_(N): the note maturity date, or the date for repayment of the    note.-   f: the fraction of the year from date t_(R) to date t_(N). This    number may depend on the day-count convention used, e.g., whether    the basis for the year is set at 365 days per year or 360 days per    year. In this example, the basis for the year is set at 360 days,    and f can be formulated as follows:

$\begin{matrix}{f = \frac{{number}\mspace{14mu} {of}\mspace{14mu} {days}\mspace{14mu} {between}\mspace{14mu} t_{R}\mspace{14mu} {and}\mspace{14mu} t_{N}}{360}} & {9.5{.1}A}\end{matrix}$

As shown in FIG. 24, the market or auction in this example is structuredsuch that the note maturity date (t_(N)) occurs on or after the optionpayout date (t_(O)) although, for example, the market auction can bestructured such that t_(N) occurs before t_(O). Additionally, asillustrated, the option payout date (t_(O)) occurs on or after the endof the observation period (t_(E)), and the end of the observation period(t_(E)) occurs on or after the premium settlement date (t_(S)). Thepremium settlement date (t_(S)), can occur on or after the end of thetrading period for the demand-based market or auction. Further, thedemand-based market or auction in this example is structured such thatthe coupon reset date (t_(R)) occurs after the premium settlement date(t_(S)) and before the note maturity date (t_(N)). However, the couponreset date (also referred to, for example, as the “FRN Fixing Date”)(t_(R)) can occur at any time before the note maturity date (t_(N)), andat any time on or after the end of the trading period or the premiumsettlement date (t_(S)). The coupon reset date (t_(R)), for example, canoccur after the end of the observation period (t_(E)) and/or the optionpayout date (t_(O)). In this example, as shown in FIG. 24, the couponreset date (t_(R)) is set between the end of the observation period(t_(E)) and the option payout date (t_(O)).

Similar to the discussion earlier in this specification in Section 1that the duration of the trading period can be unknown to theparticipants at the time that they place their orders, any of the datesabove can be pre-determined and known by the participants at the outset,or they can be unknown to the participants at the time that they placetheir orders. The end of the trading period, the premium settlement dateor the coupon reset date, for example, can occur at a randomly selectedtime, or could occur depending upon the occurrence of some eventassociated or related to the event of economic significance, or upon thefulfillment of some criterion. For example, for DBAR-enabled FRNs, thecoupon reset date could occur after a certain volume, amount, orfrequency of trading or volatility is reached in a respectivedemand-based market or auction. Alternatively, the coupon reset datecould occur, for example, after an nth catastrophic natural event (e.g.,a fourth hurricane), or after a catastrophic event of a certainmagnitude (e.g., an earthquake of a magnitude of 5.5 or higher on theRichter scale), and the natural or catastrophic event can be related orunrelated to the event of economic significance, in this example, thelevel of the ECI.

9.5.2 Variables and Formulation for Demand-Based Market or Auction

-   E: Event of economic significance, in this example, ECI. The level    of the ECI observed on t_(E). This event is the same event for the    FRN and direct digital option orders, referred to, e.g., as a    “Trigger Level” for the FRN order, and as a “strike price” for the    direct digital option order.-   L: London Interbank Offered Rate (LIBOR) from the date t_(R) to    t_(N), a variable that can be fixed, e.g., at the start of the    trading period.-   m: number of defined states, a natural number. Index letter I, i=1,    2 . . . , m. In the example shown in FIG. 9.2, for example, there    can be 7 states depending on the outcome of an economic event: the    level of the ECI on the event observation date.    -   ECI<0.7;    -   0.7<ECI<0.8;    -   0.8≦ECI<0.9;    -   0.9≦ECI<1.0;    -   1.0≦ECI<1.1;    -   1.1≦ECI<1.2; and    -   ECI≧1.2.-   n_(N): number of FRN orders in a demand-based market or auction, a    non-negative integer. Index letter j_(N), j_(N)=1,2, . . . n_(N).-   n_(D): number of direct digital option orders in a demand-based    market or auction, a non-negative integer. Index letter j_(D),    j_(D)=1,2, . . . n_(D). Direct digital option orders, include, for    example, orders which are placed using digital option parameters.-   n_(AD): number of digital option orders in an approximation set for    a j_(N) FRN order. In this example, this number is known and fixed,    e.g., at the start of the trading period, however as described    below, this number can be determined during the mapping process, a    non-negative integer. Index letter z, z=1, 2, . . . n_(AD).-   n: number of all digital option orders in a demand-based market or    auction, a non-negative integer. Index letter j, j=1, 2, . . . n.    -   The above numbers relate to one another in a single demand-based        market or auction as follows:

$\begin{matrix}{n = {{\sum\limits_{j_{N = 1}}^{n_{N}}{n_{AD}\left( j_{N} \right)}} + n_{D}}} & {9.5{.2}A}\end{matrix}$

-   L: the rate of LIBOR from date t_(R) to date t_(N)-   DF_(O): the discount factor between the premium settlement date and    the option payout date (t_(S) and t_(O)), to account for the time    value of money. DFo can be set using LIBOR (although other interest    rates may be used), and equal to, for example, 1/[1+(L* portion of    year from t_(S) to t_(O))].-   DF_(N): the discount rate between the premium settlement date and    the note maturity date, t_(S) and t_(N). DF_(N) can also be set    using LIBOR (although other interest rates may be used), and equal    to, for example, 1/[1+(L* portion of year from t_(S) to t_(N))].

9.5.3 Variables and Formulations for Each Note j_(N) in Demand-BasedMarket or Auction

-   A: notional or face amount or par of note.-   U: minimum spread to LIBOR (a positive number) specified by customer    for note, if the customer's selected outcome becomes the observed    outcome of the event. Although both buy and sell FRN orders can be    processed together with buy and sell direct digital option orders in    the same demand-based market or auction, this example demonstrates    the mapping for a buy FRN order.-   N_(P): The profit on the note if one or more of the states    corresponding to the selected outcome of the event is identified on    the event outcome date as one or more of the states corresponding to    the observed outcome (e.g., the selected outcome turns out to be the    observed outcome, or the ECI reaching or surpassing the Trigger    Level on the event outcome date), at the coupon rate, c, determined    by this demand-based market or auction.

N _(P) =A×c×f×DF _(N)  9.5.3A

-   N_(L): The loss on the note if none of the states corresponding to    the selected outcome of the event is identified on the event outcome    date as one more of the states corresponding to the observed outcome    (e.g., the selected outcome does not turn out to be the observed    outcome, or the ECI does not reach the Trigger Level on the event    outcome date).

9.5.3B

-   π: the equilibrium price of each of the digital options in the    approximation set that are filled by the demand-based market or    auction, the equilibrium price being determined by the demand-based    market or auction.    -   All of the digital options in the approximation set can have,        for example, the same payout profile or selected outcome,        matching the selected outcome of the FRN. Therefore, all of the        digital options in one approximation set that are filled by the        demand-based market or auction will have, for example, the same        equilibrium price.

9.5.4 Variables and Formulations for Each Digital Option, z, in theApproximation Set of One or More Digital Options for Each Note, j_(N),in a Demand-Based Market or Auction

-   w_(z): digital option limit price for the z^(th) digital option in    the approximation set. The digital options in the approximation set    can be arranged in descending order by limit price. The first    digital option in the set has the largest limit price. Each    subsequent digital option has a lower limit price, but the limit    price remains a positive number, such that w_(z+1)<w_(z). The number    of digital options in an approximation set can be pre-determined    before the order is placed, as in this example, or can be determined    during the mapping process as discussed below.    -   In this example, the limit price for the first digital option        (z=1) in an approximation set for one FRN order (O_(N)) can be        determined as follows:

w ₁ =DF _(O) *L/(U+L)  9.5.4A

-   -   The limit prices for subsequent digital options can be        established such that the differences between the limit prices        in the approximation set become smaller and eventually approach        zero.

-   r_(z): requested or desired payout or notional for the z^(th)    digital option in the approximation set.

-   c: coupon on the FRN, e.g., the spread to LIBOR on the FRN,    corresponding to the coupon determined after the last digital option    order in the approximation set is filled according to the    methodology discussed, for example, in Sections 6 and 7.    -   The coupon, c, can be determined, for example, by the following:

$\begin{matrix}{c = {L \times \frac{{DF}_{O} - \pi}{w_{z}}}} & {9.5{.4}B}\end{matrix}$

-   -   where w_(z) is the limit price of the last digital option order        z in the approximation set of an FRN, j_(N), to be filled by the        demand-based market or auction.

9.5.5 Formulations for the First Digital Option, z=1, in theApproximation Set of One or More Digital Options for a Note, j_(N) in aDemand-Based Market or Auction

Assuming that the first digital option in the approximation set is theonly digital option order filled by the demand-based market or auction(e.g., w₂<π≦w₁), then following equation 9.5.4B, then:

$\begin{matrix}{c = {L \times \frac{{DF}_{O} - \pi}{w_{1}}}} & {9.5{.5}A}\end{matrix}$

When the equilibrium price (for each of the filled digital options inthe approximation set) is equal to the limit price for the first digitaloption in the approximation set, π=w₁, the digital option profit is r₁(DF_(O)−w₁) and the digital option loss is r₁ w₁. Equating the option'sprofit with the note's profit yields:

r ₁(DF _(O) −w ₁)=A*U*f*DF _(N)  9.5.5B

Next, equating the option's loss with the note's loss yields:

r ₁ w ₁ =A*L*f*DF _(N)  9.5.5C

The ratio of the option's profit to the option's loss is equal to theratio of the note's profit to the note's loss:

$\begin{matrix}{\frac{r_{1}\left( {{DF}_{O} - w_{1}} \right)}{r_{1}w_{1}} = \frac{A \times U \times f \times {DF}_{N}}{A \times L \times f \times {DF}_{N}}} & {9.5{.5}D}\end{matrix}$

Simplifying this equation yields:

$\begin{matrix}{\frac{{DF}_{O} - w_{1}}{w_{1}} = \frac{U}{L}} & {9.5{.5}E} \\{\frac{{DF}_{O}}{w_{1}} = {{\frac{U}{L} + 1} = \frac{L + U}{L}}} & {9.5{.5}F}\end{matrix}$

Solving for w₁ yields:

$\begin{matrix}{w_{1} = {\left( \frac{L}{L + U} \right){DF}_{O}}} & {9.5{.5}G}\end{matrix}$

Solving for r₁ from Equation 9.5.5C yields:

r ₁ =A*L*f*DF _(N) /w ₁  9.5.5H

Substituting equation 9.5.5G for w₁ into equation 9.5.5H yields thefollowing formulation for the requested payout for the first digitaloption in the approximation set:

$\begin{matrix}{r_{1} = \frac{A \times f \times {DF}_{N} \times \left( {L + U} \right)}{{DF}_{O}}} & {9.5{.5}I}\end{matrix}$

9.5.6 Formulations for the Second Digital Option, z=2, in theApproximation Set of One or More Digital Options for a Note, j_(N), in aDemand-Based Market or Auction:

Assuming that the second digital option will be filled in theoptimization system for the entire demand-based market or auction, thecoupon earned on the note will be higher than the minimum spread toLIBOR specified by the customer, e.g., c>U.

As stated above, the profit of the FRN is A * c * f * DF_(N) and theloss if the states specified do not occur is A * L * f * DF_(N).

Now, since w₁ is determined as set forth above, and w₂ can be set assome number lower than w₁, assuming that the market or auction fillsboth the first and the second digital options and assuming that theequilibrium price is equal to the limit price for the second digitaloption (πr=w₂), the profits for the digital options if they expirein-the-money is equal to (r₁+r₂)*(DF_(O)−w₂), and the option loss isequal to (r₁+r₂)*w₂. Equating the option's profit with the note's profityields:

(r ₁ d−r ₂)(DF _(O) −w ₂)=A*c*f*DF _(N)  9.5.6A

Equating the option's loss with the note's loss yields:

(r ₁ +r ₂)w ₂ =A*L*f*DF _(N)  9.5.6B

Solving for r2 yields:

r ₂=(A*L*f*DF _(N))/w ₂ −r ₁  9.5.6C

Assuming that the second digital option is the highest order filled inthe approximation set by the demand-based market or auction, the ratioof the profits and losses of both of the options is approximately equalto the profits and losses of the FRN. This approximate equality is usedto solve for the coupon, c. Simplifying the combination of the aboveequations relating to equating the profits and losses of both options tothe profit and loss of the note, yields the following formulation forthe coupon, c, earned on the note if the note expires in-the-money andw₂>7E:

c=L*(DF _(O)−π)w ₂  9.5.6D

9.5.7 Formulations for the z^(th) Digital Option in the ApproximationSet of One or More Digital Options for a Note, j_(N), in a Demand-BasedMarket or Auction

The above description sets forth formulae involved with the first andsecond digital options in the approximation set. The following can beused to determine the requested payout for the z^(th) digital option inthe approximation set. The following can also be used as thedemand-based market's or auction's determination of a coupon for the FRNif the z^(th) digital option is the last digital option in theapproximation set filled by the demand-based market or auction (forexample, according to the optimization system discussed in Section 7),and if the FRN expires in-the-money.

The order of each digital option in the approximation set is treatedanalogously to a market order (as opposed to a limit order), where theprice of the option, it, is set equal to the limit price for the option,w_(z).

Thus, the requested payout for each digital option, r_(z), in theapproximation set can be determined according to the following formula:

$\begin{matrix}{r_{z} = {\frac{A \times L \times f \times {DF}_{N}}{w_{z}} - {\sum\limits_{x = 1}^{z - 1}r_{x}}}} & {9.5{.7}A}\end{matrix}$

Note that the determination of the requested payout for each digitaloption, r_(z), is recursively dependent on the payouts for the priordigital options, r₁, r₂, . . . , r_(z−1).

The number of digital option orders, n_(AD), used in an approximationset can be adjusted in the demand-based market or auction. For example,an FRN order could be allocated an initial set number of digital optionorders in the approximation set, and each subsequent digital optionorder could be allocated a descending limit order price as discussedabove. After these initial quantities are established for an FRN, therequested payouts for each subsequent digital option can be determinedaccording to equation 9.5.7A. If the requested payout for the z^(th)digital option in the approximation set approaches sufficiently close tozero, where z<n_(AD), then the z^(th) digital option could be set as thelast digital option needed in the approximation set, n_(AD) would thenequal z.

The coupon determined by the demand-based market or auction becomes afunction of LIBOR, the discount factor between the premium settlementdate and the option payment date, the equilibrium price, and the limitprice of the last digital option in the approximation set to be filledby the optimization system for the demand-based market or auctiondiscussed in Section 7:

c=L*(DF _(O)−π)/w _(z)  9.5.7B

where w_(z) is the limit price of the last digital option order in theapproximation set to be filled by the optimization system.

9.5.8 Numerical Example of Implementing Formulations for the z^(th)Digital Option in the Approximation Set of One or More Digital Optionsfor a Note, j_(N), in a Demand-Based Market or Auction

The following provides an illustration of a principal-protectedEmployment Cost Index-linked Floating Rate Note. In this numericalexample, the auction premium settlement date t_(S) is Oct. 24, 2001; theevent outcome date t_(E), the coupon reset date t_(R), and the optionpayout date are all Oct. 25, 2001; and the note maturity date t_(N) isJan. 25, 2002.

In this case, the discount factors can be solved using a LIBOR rate L of3.5% and a basis of Actual number of days/360:

DF_(O)=0.999903

DF_(N)=0.991135

f=0.255556

(There are 92 days of discounting between Oct. 25, 2001 and Jan. 25,2002, which is used for the computation off and DF_(N).)

The customer or note holder specifies, in this example, that the FRN isa principal protected FRN, because the principal or par or face amountor notional is paid to the note holder in the event that the FRN expiresout-of-the-money. The customer specifies the trigger level of the ECI as0.9% or higher, and the customer enters an order with a minimum spreadof 150 basis points to LIBOR. This customer will receive LIBOR plus 150bps in arrears on 100 million USD on Jan. 25, 2002, plus par if the ECIindex fixes at 0.9% or higher. This customer will receive 100 millionUSD (since the note is principal protected) on Jan. 25, 2002 if the ECIindex fixes at lower than 0.9%.

Following the notation for the variables and the formulation presentedabove, A=$100,000,000.00 (referred to as the par, principal, notional,face amount of the note) U=0.015, i.e. bidder wants to receive a minimumof 150 basis points over LIBOR

The parameters for the first digital option in the approximation set forthe demand-based market or auction are determined as follows by equation9.5.4A:

w ₁=(0.035/[0.035+0.015])*0.999903=0.70

It is reasonable to set w₂, the limit price for the second digitaloption order in the approximation set to be equal to 0.69, therefore byequation 9.5.5H:

r ₁=$100,000,000*0.035*0.255556*0.991135/0.70=$1,266,500

The coupon, c, equals 0.015 or 150 basis points, if the first digitaloption order becomes the only digital option order filled by thedemand-based market or auction and the equilibrium price is equal to thelimit price for the first digital option (π=0.7).

The parameters for the second digital option in the approximation setfor the demand-based market or auction are determined as follows,setting the limit price for this digital option to be less than thelimit price for the first digital option, or w₂=0.69, then by equation9.5.6C:

$\begin{matrix}{r_{2} = {{\$ 100}\text{,}000\text{,}000*0.035*0.255556*}} \\{{{0.991135/0.69} - {{\$ 1}\text{,}266\text{,}500}}} \\{= {{\$ 18}\text{,}306}}\end{matrix}$

If π, the equilibrium price of the digital option, is between 0.69 (w₂)and 0.70 (w₁), e.g., π=0.695, then the note coupon,c=0.0152=0.035*(0.999903−0.695)/0.70, or 152 bps spread to LIBOR byequation 9.5.5A. This becomes the coupon for the note if thedemand-based market or auction only fills the first digital option orderin the approximation set and if the demand-based market or auction setsthe equilibrium price for the selected outcome equal to 0.695.

If π, the equilibrium price of the digital option, is equal to 0.69(w₂), the coupon for the FRN becomes 157 basis points if the seconddigital option is the highest digital option order filled by thedemand-based market or auction, by equation 9.5.6D:

c=0.035*(0.999903−0.69)/0.69=0.0157 or 157 basis points

The requested payouts for each subsequent digital option, and thesubsequently determined coupon on the note (determined pursuant to thelimit price of the last digital option in the approximation set to befilled by the demand-based market or auction and the equilibrium pricefor the selected outcome), are determined using equations 9.5.7A and9.5.7B.

9.6 CONCLUSION

These equations present one example of how to map FRNs and swaps intoapproximation sets comprised of digital options, transforming these FRNsand swaps into DBAR-enabled FRNs and swaps. The mapping can occur at aninterface in a demand-based market or auction, enabling otherwisestructured instruments to be evaluated and traded alongside digitaloptions, for example, in the same optimization solution. As shown inFIG. 25, the methods in this embodiment can be used to createDBAR-enabled products out of any structured instruments, so that avariety of structured instruments and digital options can be traded andevaluated in the same efficient and liquid demand-based market orauction, thus significantly expanding the potential size of demand-basedmarkets or auctions.

10. DETAILED DESCRIPTION OF THE DRAWINGS

Referring now to the drawings, similar components appearing in differentdrawings are identified by the same reference numbers.

FIGS. 1 and 2 show schematically a preferred embodiment of a networkarchitecture for any of the embodiments of a demand-based market orauction or DBAR contingent claims exchange (including digital options).As depicted in FIG. 1 and FIG. 2, the architecture conforms to adistributed Internet-based architecture using object oriented principlesuseful in carrying out the methods of the present invention.

In FIG. 1, a central controller 100 has a plurality software andhardware components and is embodied as a mainframe computer or aplurality of workstations. The central controller 100 is preferablylocated in a facility that has back-up power, disaster-recoverycapabilities, and other similar infrastructure, and is connected viatelecommunications links 110 with computers and devices 160, 170, 180,190, and 200 of traders and investors in groups of DBAR contingentclaims of the present invention. Signals transmitted usingtelecommunications links 110, can be encrypted using such algorithms asBlowfish and other forms of public and private key encryption. Thetelecommunications links 110 can be a dialup connection via a standardmodem 120; a dedicated line connection establishing a local area network(LAN) or wide area network (WAN) 130 running, for example, the Ethernetnetwork protocol; a public Internet connection 140; or wireless orcellular connection 150. Any of the computers and devices 160, 170, 180,190 and 200, depicted in FIG. 1, can be connected using any of the links120, 130, 140 and 150 as depicted in hub 111. Other telecommunicationslinks, such as radio transmission, are known to those of skill in theart.

As depicted in FIG. 1, to establish telecommunications connections withthe central controller 100, a trader or investor can use workstations160 running, for example, UNIX, Windows NT, Linux, or other operatingsystems. In preferred embodiments, the computers used by traders orinvestors include basic input/output capability, can include a harddrive or other mass storage device, a central processor (e.g., anIntel-made Pentium III processor), random-access memory, networkinterface cards, and telecommunications access. A trader or investor canalso use a mobile laptop computer 180, or network computer 190 having,for example, minimal memory and storage capability 190, or personaldigital assistant 200 such as a Palm Pilot. Cellular phones or othernetwork devices may also be used to process and display information fromand communicate with the central controller 100.

FIG. 2 depicts a preferred embodiment of the central controller 100comprising a plurality of software and hardware components. Computerscomprising the central controller 100 are preferably high-endworkstations with resources capable of running business operatingsystems and applications, such as UNIX, Windows NT, SQL Server, andTransaction Server. In a preferred embodiment, these computers arehigh-end personal computers with Intel-made (x86 “instruction set”)CPUs, at least 128 megabytes of RAM, and several gigabytes of hard drivedata storage space. In preferred embodiments, computers depicted in FIG.2 are equipped with JAVA virtual machines, thereby enabling theprocessing of JAVA instructions. Other preferred embodiments of thecentral controller 100 may not require the use of JAVA instruction sets.

In a preferred embodiment of central controller 100 depicted in FIG. 2,a workstation software application server 210, such as the WeblogicServer available from BEA Systems, receives information viatelecommunications links 110 from investors' computers and devices 160,170, 180, 190 and 200. The software application server 210 isresponsible for presenting human-readable user interfaces to investors'computers and devices, for processing requests for services frominvestors' computers and devices, and for routing the requests forservices to other hardware and software components in the centralcontroller 100. The user interfaces that can be available on thesoftware application server 210 include hypertext markup language (HTML)pages, JAVA applets and servlets, JAVA or Active Server pages, or otherforms of network-based graphical user interfaces known to those of skillin the art. For example, investors or traders connected via an Internetconnection for HTML can submit requests to the software applicationserver 210 via the Remote Method Invocation (RMI) and/or the InternetInter-Orb Protocol (HOP) running on top of the standard TCP/IP protocol.Other methods are known to those of skill in the art for transmittinginvestors' requests and instructions and presenting human readableinterfaces from the application server 210 to the traders and investors.For example, the software application server 210 may host Active ServerPages and communicate with traders and investors using DCOM.

In a preferred embodiment, the user interfaces deployed by the softwareapplication server 210 present login, account management, trading,market data, and other input/output information necessary for theoperation of a system for investing in groups of DBAR contingent claimsaccording to the present invention. A preferred embodiment uses the HTMLand JAVA appletlservlet interface. The HTML pages can be supplementedwith embedded applications or “applets” using JAVA based or ActiveXstandards or another suitable application, as known to one of skill inthe art.

In a preferred embodiment, the software application server 210 relies onnetwork-connected services with other computers within the centralcontroller 100. The computers comprising the central controller 100preferably reside on the same local area network (e.g., Ethernet LAN)but can be remotely connected over Internet, dedicated, dialup, or othersimilar connections. In preferred embodiments, networkintercommunication among the computers comprising central controller 100can be implemented using DCOM, CORBA, or TCP/IP or other stack servicesknown to those of skilled in the art.

Representative requests for services from the investors' computers tothe software application server 210 include: (1) requests for HTML pages(e.g., navigating and searching a web site); (2) logging onto the systemfor trading DBAR contingent claims; (3) viewing real-time and historicalmarket data and market news; (4) requesting analytical calculations suchas returns, market risk, and credit risk; (5) choosing a group of DBARcontingent claims of interest by navigating HTML pages and activatingJAVA applets; (6) making an investment in one or more defined states ofa group of DBAR contingent claims; and (7) monitoring investments ingroups of DBAR contingent claims.

In a preferred embodiment depicted in FIG. 2, an Object Request Broker(ORB) 230 can be a workstation computer operating specialized softwarefor receiving, aggregating, and marshalling service requests from thesoftware application server 210. For example, the ORB 230 can operate asoftware product called Visibroker, available from Inprise, and relatedsoftware products that provide a number of functions and servicesaccording to the Common Object Request Broker Architecture (CORBA)standard. In a preferred embodiment, one function of the ORB 230 is toprovide what are commonly known in the object-oriented software industryas directory services, which correlate computer code organized intoclass modules, known as “objects,” with names used to access thoseobjects. When an object is accessed in the form of a request by name,the object is instantiated (i.e., caused to run) by the ORB 230. Forexample, in a preferred embodiment, computer code organized into a JAVAclass module for the purpose of computing returns using a canonical DRFis an object named “DRF_Returns,” and the directory services of the ORB230 would be responsible for invoking this object by this name wheneverthe application server 210 issues a request that returns be computed.Similarly, in the case of DBAR digital options, computer code organizedinto a JAVA class module for the purpose of computing investment amountsusing a canonical DRF is an object named “OPF_Prices,” and the directoryservices of the ORB 230 would also be responsible for invoking thisobject by this name whenever the application server 210 issues a requestthat prices or investment amounts be computed.

In a preferred embodiment, another function of the ORB 230 is tomaintain what is commonly known in the object-oriented software industryas an interface repository, which contains a database of objectinterfaces. The object interfaces contain information regarding whichcode modules perform which functions. For example, in a preferredembodiment, a part of the interface of the object named “DRF_Returns” isa function which fetches the amount currently invested across thedistribution of states for a group of DBAR contingent claims. Similarly,for DBAR digital options, a part of the interface of the object named“OPF_Prices” is a function which fetches the requested payout orreturns, the selected outcomes and the limit prices or amounts for eachin a group of DBAR digital options.

In a preferred embodiment, as in the other embodiments of the presentinvention, another function of the ORB 230 is to manage the length ofruntime for objects which are instantiated by the ORB 230, and to manageother functions such as whether the objects are shared and how theobjects manage memory. For example, in a preferred embodiment, the ORB230 determines, depending upon the request from the software applicationserver 210, whether an object which processes market data will sharesuch data with other objects, such as objects that allocate returns toinvestments in defined states.

In a preferred embodiment, as the other embodiments of the presentinvention, another function of the ORB 230 is to provide the ability forobjects to communicate asynchronously by responding to messages or dataat varying times and frequencies based upon the activity of otherobjects. For example, in a preferred embodiment, an object that computesreturns for a group of DBAR contingent claims responds asynchronously inreal-time to a new investment and recalculates returns automaticallywithout a request by the software application server 210 or any otherobject. In preferred embodiments, such asynchronous processes areimportant where computations in real-time are made in response to otheractivity in the system, such as a trader making a new investment or thefulfillment of the predetermined termination criteria for a group ofDBAR contingent claims.

In a preferred embodiment, as the other embodiments of the presentinvention, another function of the ORB 230 is to provide functionsrelated to what is commonly known in the object-oriented softwareindustry as marshalling. Marshalling in general is the process ofobtaining for an object the relevant data it needs to perform itsdesignated function. In preferred embodiments of the present invention,such data includes for example, trader and account information and canitself be manipulated in the form of an object, as is common in thepractice of object-oriented programming. Other functions and servicesmay be provided by the ORB 230, such as the functions and servicesprovided by the Visibroker product, according to the standards andpractices of the object-oriented software industry or as known to thoseof skill in the art.

In a preferred embodiment depicted in FIG. 2, which can be applied tothe other embodiments of the present invention, transaction server 240is a computer running specialized software for performing various tasksincluding: (1) responding to dka requests from the ORB 230, e.g., user,account, trade data and market data requests; (2) performing relevantcomputations concerning groups of DBAR contingent claims, such asintra-trading period and end-of-trading-period returns allocations andcredit risk exposures; and (3) updating investor accounts based upon DRFpayouts for groups of DBAR contingent claims and applying debits orcredits for trader margin and positive outstanding investment balances.The transaction server 240 preferably processes all requests from theORB 230 and, for those requests that require stored data (e.g., investorand account information), queries data storage devices 260. In apreferred embodiment depicted in FIG. 2, a market data feed 270 suppliesreal-time and historical market data, market news, and corporate actiondata, for the purpose of ascertaining event outcomes and updatingtrading period returns. The specialized software running on transactionserver 240 preferably incorporates the use of object oriented techniquesand principles available with computer languages such as C++ or Java forimplementing the above-listed tasks.

As depicted in FIG. 2, in a preferred embodiment the data storagedevices 260 can operate relational database software such as Microsoft'sSQL Server or Oracle's 8i Enterprise Server. The types of databaseswithin the data storage devices 260 that can be used to support the DBARcontingent claim and exchange preferably comprise: (1) Trader andAccount databases 261; (2) Market Returns databases 262; (3) Market Datadatabases 263; (4) Event Data databases 264; (5) Risk databases 265; (6)Trade Blotter databases 266; and (7) Contingent Claims Terms andConditions databases 267. The kinds of data preferably stored in eachdatabase are shown in more detail in FIG. 4. In a preferred embodiment,connectivity between data storage devices 260 and transaction server 240is accomplished via TCP/IP and standard Database Connectivity Protocols(DBC) such as the JAVA DBC (JDBC). Other systems and protocols for suchconnectivity are known to those of skill in the art.

In reference to FIG. 2, application server 210 and ORB 230 may beconsidered to form an interface processor, while transaction server 240forms a demand-based transaction processor. Further, the databaseshosted on data storage devices 260 may be considered to form a tradestatus database. Investors, also referred to as traders, communicatingvia telecommunications links 110 from computers and devices 160, 170,180, 190, and 200, may be considered to perform a series of demand-basedinteractions, also referred to as demand-based transactions, with thedemand-based transaction processor. A series of demand-basedtransactions may be used by a trader, for example, to obtain marketdata, to establish a trade, or to close out a trade.

FIG. 3 depicts a preferred embodiment of the implementation of a groupof DBAR contingent claims. As depicted in FIG. 3, an exchange firstselects an event of economic significance 300. In the preferredembodiment, the exchange then partitions the possible outcomes for theevent into mutually exclusive and collectively exhaustive states 305,such that one state among the possible states in the partitioneddistribution is guaranteed to occur, and the sum of probabilities of theoccurrence of each partitioned state is unity. Trading can then commencewith the beginning 311 of the first trading period 310. In the preferredembodiment depicted in FIG. 3, a group of DBAR contingent claims hastrading periods 310, 320, 330, and 340, with trading period start date311, 321, 331, 341 respectively, followed by a predetermined timeinterval by each trading period's respective trading end dates 313, 323,333 and 343. The predetermined time interval is preferably of shortduration in order to attain continuity. In the preferred embodiment,during each trading period the transaction server 240 running JAVA codeimplementing the DRF for the group of DBAR contingent claims adjustsreturns immediately in response to changes in the amounts invested ineach of the defined states. Changes in market conditions during atrading period, such as price and volatility changes, as well as changesin investor risk preferences and liquidity conditions in the underlyingmarket, among other factors, will cause amounts invested in each definedstate to change thereby reflecting changes in expectations of tradersover the distribution of states defining the group of DBAR contingentclaims.

In a preferred embodiment, the adjusted returns calculated during atrading period, i.e., intra-trading period returns, are of informationalvalue only—only the returns which are finalized at the end of eachtrading period are used to allocate gains and losses for a trader'sinvestments in a group or portfolio of groups of DBAR contingent claims.In a preferred embodiment, at the end of each trading period, forexample, at trading end dates 313, 323, 333, and 343, finalized returnsare allocated and locked in. The finalized returns are the rates ofreturn to be allocated per unit of amount invested in each defined stateshould that state occur. In a preferred embodiment, each trading periodcan therefore have a different set of finalized returns as marketconditions change, thereby enabling traders to make investments duringlater trading periods which hedge investments from earlier tradingperiods that have since closed.

In another preferred embodiment, not depicted, trading periods overlapso that more than one trading period is open for investment on the sameset of predefined states. For example, an earlier trading period canremain open while a subsequent trading period opens and closes. Otherpermutations of overlapping trading periods are possible and areapparent to one of skill in the art from this specification or practiceof the present invention.

The canonical DRF, as previously described, is a preferred embodiment ofa DRF which takes investment across the distribution of states and eachstate, the transaction fee, and the event outcome and allocates a returnfor each state if it should occur. A canonical DRF of the presentinvention, as previously described, reallocates all amounts invested instates that did not occur to the state that did occur. Each trader thathas made an investment in the state that did occur receives a pro-ratashare of the trades from the non-occurring states in addition to theamount he originally invested in the occurring state, less the exchangefee.

In the preferred embodiment depicted in FIG. 3, at the close of thefinal trading period 343, trading ceases and the outcome for the eventunderlying the contingent claim is determined at close of observationperiod 350. In a preferred embodiment, only the outcome of the eventunderlying the group of contingent claims must be uncertain during thetrading periods while returns are being locked in. In other words, theevent underlying the contingent claims may actually have occurred beforethe end of trading so long as the actual outcome remains unknown, forexample, due to the time lag in measuring or ascertaining the event'soutcome. This could be the case, for instance, with macroeconomicstatistics like consumer price inflation.

In the preferred embodiment depicted in FIG. 2, once the outcome isobserved at time 350, process 360 operates on the finalized returns fromall the trading periods and determines the payouts. In the case of acanonical DRF previously described, the amounts invested in the losinginvestments finance the payouts to the successful investments, less theexchange fee. In a canonical DRF, successful investments are those madeduring a trading period in a state which occurred as determined at time350, and unsuccessful investments are those made in states which did notoccur. Examples 3.1.1-3.1.25 above illustrate various preferredembodiments of a group of DBAR contingent claims using a canonical DRF.In the preferred embodiment depicted in FIG. 3, the results of process360 are made available to traders by posting the results for all tradingperiods on display 370. In a preferred embodiment not depicted, traderaccounts are subsequently updated to reflect these results.

FIG. 4 provides a more detailed depiction of the data storage devices260 of a preferred embodiment of a DBAR contingent claims exchange whichcan be applied to the other embodiments of the present invention. In apreferred embodiment, data storage devices 260, on which relationaldatabase software is installed as described above, is a non-volatilehard drive data storage system, which may comprise a single device ormedium, or may be distributed across a plurality of physical devices,such as a cluster of workstation computers operating relational databasesoftware, as described previously and as known in the art. In apreferred embodiment, the relational database software operating on thedata storage devices 260 comprises relational database tables, storedprocedures, and other database entities and objects that are commonlycontained in relational database software packages. In the preferredembodiment depicted in FIG. 4, databases 261-267 each contain suchtables and other relational database entities and objects necessary ordesirable to implement an embodiment of the present invention. FIG. 4identifies the kinds of information that can be stored in such devices.Of course, the kinds of data shown in the drawing are not exhaustive.The storage of other data on the same or additional databases may beuseful depending on the nature of the contingent claim being traded.Moreover, in the preferred embodiment depicted in FIG. 4, certain dataare shown in FIG. 4 as stored in more than one storage device. Invarious other preferred embodiments, such data may be stored in only onesuch device or may be calculated. Other database designs andarchitectures will be apparent to those of skill in the art from thisspecification or practice of the present invention.

In the preferred embodiment depicted in FIG. 4, the Trader and Accountdatabase 261 stores data related to the identification of a DBAR tradersuch as name, password, address, trader identification number, etc. Datarelated to the trader's credit rating can also be stored and updated inresponse to changes in the trader's credit status. Other informationthat can be stored in Trader and Account database 261 includes datarelated to the trader's account, for example, active and inactiveinvestments, the trader's balance, the trader's margin limits,outstanding margin amounts, interest credited on outstanding tradebalances and interest paid on outstanding margin balances, anyrestrictions the trader may have regarding access to his account, andthe trader's profit and loss information regarding active and inactiveinvestments. Information related to multi-state investments to beallocated can also be stored in Trader and Account database 261. Thedata stored in database 261 can be used, for example, to issue accountrelated statements to traders.

In the preferred embodiment depicted in FIG. 4, the Market Returnsdatabase 262 contains information related to returns available atvarious times for active and inactive groups of DBAR contingent claims.In a preferred embodiment, each group of contingent claims in database262 is identified using a unique identifier previously assigned to thatgroup. Returns for each defined state for each group of contingentclaims reflected are stored in database 262. Returns calculated andavailable for display to traders during a given trading period arestored in database 262 for each state and for each claim. At the end ofeach trading period, finalized returns are computed and stored in MarketReturns database 262. Marginal returns, as previously described, canalso be stored in database 262. The data in Market Returns database 262may also include information relevant to a trader's decisions such ascurrent and past intra-period returns, as well as information used todetermine payouts by a DRF or investment amounts by an OPF for a groupof DBAR contingent claims.

In the preferred embodiment depicted in FIG. 4, Market Data database 263stores market data from market data feed 270. In a preferred embodiment,the data in Market Data database 263 include data relevant for the typesof contingent claims that can be traded on a particular exchange. In apreferred embodiment, real-time market data include data such asreal-time prices, yields, index levels, and other similar information.

In a preferred embodiment, such real-time data from Market Data database263 are presented to traders to aid in making investment decisions canbe used by the DRF to allocate returns and by the OPF to determineinvestment amounts for groups of contingent claims that depend on suchinformation. Historical data relating to relevant groups of DBARcontingent claims can also be stored in Market Data database 263. Inpreferred embodiments, news items related to underlying groups of DBARcontingent claims (e.g., comments by the Federal Reserve) are alsostored in Market Data database 263 and can be retrieved by traders.

In the preferred embodiment depicted in FIG. 4, Event Data database 264stores data related to events underlying the groups of DBAR contingentclaims that can be traded on an exchange. In a preferred embodiment,each event is identified by a previously assigned event identificationnumber. Each event has one or more associated group of DBAR contingentclaims based on that event and is so identified with a previouslyassigned contingent claim group identification number. The type of eventcan also be stored in Event database 264, for example, whether the eventis based on a closing price of a security, a corporate earningsannouncement, a previously calculated but yet to be released economicstatistic, etc. The source of data used to determine the outcome of theevent can also be stored in Event database 264. After an event outcomebecomes known, it can also be stored in Event database 264 along withthe defined state of the respective group of contingent claimscorresponding to that outcome.

In the preferred embodiment depicted in FIG. 4, Risk database 265 storesthe data and results and analyses related to the estimation andcalculation of market risk and credit risk. In a preferred embodiment,Risk database 265 correlates the derived results with an accountidentification number. The market and credit risk quantities that can bestored are those related to the calculation of CAR and CCAR, such as thestandard deviation of unit returns for each state, the standarddeviation of dollar returns for each state, the standard deviation ofdollar returns for a given contingent claim, and portfolio CAR.Intermediate estimation and simulation data such as correlation matricesused in VAR-based CAR and CCAR calculations and scenarios used inMCS-based calculations can also be stored in Risk database 265.

In the preferred embodiment depicted in FIG. 4, Trade Blotter database266 contains data related to the investments, both active and inactive,made by traders for all the groups of DBAR contingent claims that can betraded on the particular exchange. Such data may include previouslyassigned trader identification numbers previously assigned investmentidentification numbers, previously assigned account identificationnumbers, previously assigned contingent claim identification numbers,state identification numbers previously assigned corresponding to eachdefined state, the time of each investment, the units of value used tomake each investments (e.g., dollars), the investment amounts, thedesired or requested payouts or returns, the limits on investmentamounts (for DBAR digital options), how much margin is used to make theinvestments, and previously assigned trading period identificationnumbers. In addition, data related to whether an investment is amulti-state investment can also be stored. The payout distribution thata trader desires to replicate and that the exchange will implement usinga multi-state investment allocation, as described above, can also bestored in Trade Blotter database 266.

In the preferred embodiment depicted in FIG. 4, Contingent Claims Termsand Conditions database 267 stores data related to the definition andstructure of each group of DBAR contingent claims. In a preferredembodiment, such data are called “terms and conditions” to indicate thatthey relate to the contractual terms and conditions under which tradersagree to be bound, and roughly correspond to material found inprospectuses in traditional markets. In a preferred embodiment, as wellas other embodiments of the present invention, the terms and conditionsprovide the fundamental information regarding the nature of thecontingent claim to be traded, e.g., the number of trading periods, thetrading period(s)' start and end times, the type of event underlying thecontingent claim, how the DRF finances successful investments fromunsuccessful investments, how the OPF determines order prices orinvestment amounts as a function of the requested payout, selection ofoutcomes and limits for each order for a DBAR digital options auction ormarket, the time at which the event is observed for determining theoutcome, other predetermined termination criteria, the partition ofstates in which investments can be made, and the investment and payoutvalue units (e.g., dollars, numbers of shares, ounces of gold, etc.). Ina preferred embodiment, contingent claim and event identificationnumbers are assigned and stored in Contingent Claims Terms andConditions database 267 so that they may be readily referred to in othertables of the data storage devices.

FIG. 5 shows a flow diagram depicting illustrative processes used andillustrative decisions made by a trader using a preferred embodiment ofthe present invention. For purposes of illustration in FIG. 5, it isassumed that the trader is making an investment in a DBAR rangederivative (RD) examples of which are disclosed above. In particular, itis assumed for the purposes of illustration that the DBAR RD investmentbeing made is in a contingent claim based upon the closing price of IBMcommon stock on Aug. 3, 1999 (as indicated in the display 501 of FIG.6).

In process 401, depicted in FIG. 5, the trader requests access to theDBAR contingent claim exchange. As previously described in a preferredembodiment, the software application server 210 (depicted in FIG. 2)processes this request and routes it to the ORB 230, which instantiatesan object responsible for the authentication of traders on the exchangeon transaction server 240. The authentication object on transactionserver 240, for example, queries the Trader and Account database 261(depicted in FIG. 4) for the trader's username, password, and otheridentifying information supplied. The authentication object responds byeither allowing or denying access as indicated in process 402 depictedin FIG. 5. If authentication fails in this illustration, process 403prompts the trader to retry a logon or establish valid credentials forlogging onto the system. If the trader has been granted access, thesoftware application server 210 (depicted in FIG. 2) will display to thetrader many user interfaces that may be of interest. For example, in apreferred embodiment, the trader can navigate through a sample of groupsof DBAR contingent claims currently being traded, as represented inprocess 404.

The trader may also check current market conditions by requesting thoseinterfaces in process 404 that contain current market data as obtainedfrom market data feed 270 (depicted in FIG. 2) and stored in Market Datadatabase 263 (as depicted in FIG. 4). Process 405 of FIG. 5 representsthe trader requesting the application server 210 for relevantinformation regarding the trader's account, such as the trader's currentportfolio of trades, trade amounts, current amount of marginoutstanding, and account balances. In a preferred embodiment, thisinformation is obtained by objects running on transaction server 240(FIG. 2) that query Trader and Account database 261 and Trade Blotterdatabase 266 (FIG. 4).

As depicted in FIG. 5, process 407 represents the selection of a groupof DBAR contingent claims by a trader for the purpose of making aninvestment. The application server 210 (depicted in FIG. 2) can presentuser interfaces to the trader such as the interface shown in FIG. 6 asis known in the art. Process 408 represents the trader requesting dataand analysis which may include calculations as to the effect thetrader's proposed investment would have on the current returns. Thecalculations can be made using the implied “bid” and “offer” demandresponse equations described above. The processes that perform thesedata requests and manipulation of such data are, in a preferredembodiment, objects running on transaction server 240 (as depicted inFIG. 2). These objects, for example, obtain data from database 262 (FIG.4) by issuing a query that requests investment amounts across thedistribution of states for a given trading period for a given group ofcontingent claims. With the investment amount data, other objectsrunning on transaction server 240 (FIG. 2) can perform marginal returnscalculations using the DRF of the group of contingent claims asdescribed above. Such processes are objects managed by the ORB 230 (asdepicted in FIG. 2).

Returning to the illustration depicted in FIG. 5, process 411 representsa trader's decision to make an investment for a given amount in one ormore defined states of the group of DBAR contingent claims of interest.In a preferred embodiment, the trader's request to make an investmentidentifies the particular group of claims, the state or states in whichinvestments are to be made, the amount to be invested in the state orstates, and the amount of margin to be used, if any, for theinvestments.

Process 412 responds to any requests to make an investment on margin.The use of margin presents the risk that the exchange may not be able tocollect the entire amount of a losing investment. Therefore, inpreferred embodiments, an analysis is performed to determine the amountof risk to which a current trader is exposed in relation to the amountof margin loans the trader currently has outstanding. In process 413such an analysis is carried out in response to a margin request by thetrader.

The proposed trade or trades under consideration may have the effect ofhedging or reducing the total amount of risk associated with thetrader's active portfolio of investments in groups of DBAR contingentclaims. Accordingly, in a preferred embodiment, the proposed trades andmargin amounts should be included in a CAR analysis of the trader'sportfolio.

In a preferred embodiment, the CAR analysis performed by process 413,depicted in FIG. 5, can be conducted according to the VAR, MCS, or HSmethodologies previously discussed, using data stored in Risk database265 (FIG. 2), such as correlation of state returns, correlation ofunderlying events, etc. In a preferred embodiment, the results of theCAR calculation are also stored in Risk database 265. As depicted inFIG. 5, process 414 determines whether the trader has sufficient equitycapital in his account by comparing the computed CAR value and thetrader's equity in accordance with the exchange's margin rules. Inpreferred embodiments, the exchange requires that all traders maintain alevel of equity capital equal to some portion or multiple of the CARvalue for their portfolios. For example, assuming CAR is computed with a95% statistical confidence as described above, the exchange may requirethat traders have 10 times CAR as equity in their accounts. Such arequirement would mean that traders would suffer drawdowns to equity of10% approximately 5% of the time, which might be regarded as areasonable tradeoff between the benefits of extending margin to tradersto increase liquidity and the risks and costs associated with traderdefault. In addition, in preferred embodiments, the exchange can alsoperform CCAR calculations to determine the amount of credit risk in thegroup of DBAR contingent claims due to each trader. In a preferredembodiment, if a trader does not have adequate equity in his account orthe amount of credit risk posed by the trader is too great, the requestfor margin is denied, as depicted in process 432 (FIG. 5).

As further depicted in FIG. 5, if the trader has requested no margin orthe trader has passed the margin tests applied in process 414, process415 determines whether the investment is one to be made over multiplestates simultaneously in order to replicate a trader's desired payoutdistribution over such states. If the investment is multi-state, process460 requests trader to enter a desired payout distribution. Suchcommunication will comprise, for example, a list of constituent statesand desired payouts in the event that each constituent state occurs. Forexample, for a four-state group of DBAR contingent claims, the tradermight submit the four dimensional vector (10, 0, 5, 2) indicating thatthe trader would like to replicate a payout of 10 value units (e.g.,dollars) should state 1 occur, no payout should state 2 occur, 5 unitsshould state 3 occur, and 2 units should state 4 occur. In a preferredembodiment, this information is stored in Trade Blotter database 266(FIG. 4) where it will be available for the purposes of determining theinvestment amounts to be allocated among the constituent states for thepurposed of replicating the desired payouts. As depicted in FIG. 5, ifthe investment is a multi-state investment, process 417 makes aprovisional allocation of the proposed investment amount to each of theconstituent states.

As further depicted in FIG. 5, the investment details and information(e.g., contingent claim, investment amount, selected state, amount ofmargin, provisional allocation, etc.) are then displayed to the traderfor confirmation by process 416. Process 418 represents the trader'sdecision whether to make the investment as displayed. If the traderdecides against making the investment, it is not executed as representedby process 419. If the trader decides to make the investment and process420 determines that it is not a multi-state investment, the investmentis executed, and the trader's investment amount is recorded in therelevant defined state of the group of DBAR contingent claims accordingto the investment details previously accepted. In a preferredembodiment, the Trade Blotter database 266 (FIG. 4) is then updated byprocess 421 with the new investment information such as the trader ID,trade ID, account identification, the state or states in whichinvestments were made, the investment time, the amount invested, thecontingent claim identification, etc.

In the illustration depicted in FIG. 5, if the trader decides to makethe investment, and process 420 determines that it is a multi-stateinvestment, process 423 allocates the invested amount to the constituentstates comprising the multi-state investment in amounts that generatethe trader's desired payout distribution previously communicated to theexchange in process 460 and stored in Trader Blotter database 266 (FIG.4). For example, in a preferred embodiment, if the desired payouts areidentical payouts no matter which state occurs among the constituentstates, process 423 will update a suspense account entry and allocatethe multi-state trade in proportion to the amounts previously investedin the constituent states. Given the payout distribution previouslystored, the total amount to be invested, and the constituent states inwhich the “new” investment is to be made, then the amount to be investedin each constituent state can be calculated using the matrix formulaprovided in Example 3.1.21, for example. Since these calculations dependon the existing distributions of amounts invested both during and at theend of trading, in a preferred embodiment reallocations are performedwhenever the distribution of amounts invested (and hence returns)change.

As further depicted in FIG. 5, in response to a new investment, Process422 updates the returns for each state to reflect the new distributionof amounts invested across the defined states for the relevant group ofDBAR contingent claims. In particular, process 422 receives the newtrade information from Trade Blotter database 266 as updated by process421, if the investment is not multi-state, or from Trader and Accountdatabase 261 as updated by suspense account process 423, if theinvestment is a multi-state investment. Process 422 involves the ORB 230(FIG. 2) instantiating an object on transaction server 240 forcalculating returns in response to new trades. In this illustration, theobject queries the new trade data from the Trade Blotter database 266 orthe suspense account in Trader and Account database 261 (FIG. 4),computes the new returns using the DRF for the group of contingentclaims, and updates the intra-trading period returns stored in MarketReturns database 262.

As depicted in FIG. 5, if the investment is a multi-state investment asdetermined by process 450, the exchange continues to update the suspenseaccount to reflects the trader's desired payout distribution in responseto subsequent investments entering the exchange. Any updatedintra-trading period returns obtained from process 422 and stored inMarket Returns database 262 are used by process 423 to perform areallocation of multi-state investments to reflect the updated returns.If the trading period has not closed, as determined by process 452, thereallocated amounts obtained from the process 423 are used, along withinformation then simultaneously stored in Trade Blotter database 266(FIG. 4), to perform further intra-trading period update of returns, perprocess 422 shown in FIG. 5. However, if the trading period has closed,as determined in this illustration by process 452, then the multi-statereallocation is performed by process 425 so that the returns for thetrading period can be finalized per process 426.

In a preferred embodiment, the closing of the trading period is animportant point since at that point the DRF object running onTransaction server 240 (FIG. 2) calculates the finalized returns andthen updates Market Returns database 262 with those finalized returns,as represented by process 426 depicted in FIG. 5. The finalized returnsare those which are used to compute payouts once the outcome of theevent and, therefore, the state which occurred are known and all otherpredetermined termination criteria are fulfilled. Even though amulti-state reallocation process 425 is shown in FIG. 5 between process452 and process 426, multi-state reallocation process 425 is not carriedout if the investment is not a multi-state investment.

Continuing with the illustration depicted in FIG. 5, process 427represents the possible existence of subsequent trading periods for thesame event on which the given group of DBAR contingent claims is based.If such periods exist, traders may make investments during them, andeach subsequent trading period would have its own distinct set offinalized returns. For example, the trader in a group of contingentclaims may place a hedging investment in one or more of the subsequenttrading periods in response to changes in returns across the tradingperiods in accordance with the method discussed in Example 3.1.19 above.The ability to place hedging trades in successive trading periods, eachperiod having its own set of finalized returns, allows the trader tolock-in or realize profits and losses in virtually continuous time asreturns change across the trading periods. In a preferred embodiment,the plurality of steps represented by process 427 are performed aspreviously described for the earlier portions of FIG. 5.

As further depicted in FIG. 5, process 428 marks the end of all thetrading periods for a group of contingent claims. In a preferredembodiment, at the end of the last trading period, the Market Returnsdatabase 262 (FIG. 4) contains a set of finalized returns for eachtrading period of the group of contingent claims, and Trade Blotterdatabase 266 contains data on every investment made by every trader onthe illustrative group of DBAR contingent claims.

In FIG. 5, process 429 represents the observation period during whichthe outcome of the event underlying the contingent claim is observed,the occurring state of the DBAR contingent claim determined and anyother predetermined termination criteria are fulfilled. In a preferredembodiment, the event outcome is determined by query of the Market Datadatabase 263 (FIG. 4), which has been kept current by Market Data Feed270. For example, for a group of contingent claims on the event of theclosing price of IBM on Aug. 3, 1999, the Market Data database 263 willcontain the closing price, 119⅜, as obtained from the specified eventdata source in Event Data database 264. The event data source might beBloomberg, in which case an object residing on transaction server 240previously instantiated by ORB 230 will have updated the Market Returnsdatabase 262 with the closing price from Bloomberg. Another similarlyinstantiated object on transaction server 240 will query the MarketReturns database 262 for the event outcome (119⅜), will query theContingent Claims Terms and Conditions database 267 for the purpose ofdetermining the state identification corresponding to the event outcome(e.g., Contingent Claim # 1458, state #8) and update the event and stateoutcomes into the Event Data database 264.

As further depicted in FIG. 5, process 430 shows an object instantiatedon transaction server 240 by ORB 230 performing payout calculations inaccordance with the DRF and other terms and conditions as contained inContingent Claims Terms and Conditions database 267 for the given groupof contingent claims. In a preferred embodiment, the object isresponsible for calculating amounts to be paid to successful investmentsand amounts to be collected from unsuccessful investments, i.e.,investments in the occurring and non-occurring states, respectively.

As further depicted in FIG. 5, process 431 shows trader account datastored in Trader and Account database 261 (FIG. 4) being updated by theobject which determines the payouts in process 430. Additionally, inprocess 431 in this illustration and preferred embodiments, outstandingcredit and debit interest corresponding to positive and margin balancesare applied to the relevant accounts in Trader and Account database 261.

FIG. 6 depicts as preferred embodiment of a sample HTML page used bytraders in an exchange for groups of DBAR contingent claims whichillustrates sample display 500 with associated input/output devices,such as display buttons 504-507 and can be used with other embodimentsof the present invention. As depicted in FIG. 6, descriptive data 501illustrate the basic investment and market information relevant to aninvestment. In the investment illustrated in FIG. 6, the event is theclosing price of IBM common stock at 4:00 p.m. on Aug. 3, 1999. Asdepicted in FIG. 6, the sample HTML page displays amount invested ineach defined state, and returns available from Market Returns database262 depicted in FIG. 4. In this illustration and in preferredembodiments, returns are calculated on transaction server 240 (FIG. 2)using, for example, a canonical DRF. As also depicted in FIG. 6,real-time market data is displayed in an intraday “tick chart”,represented by display 503, using data obtained from Market Data Feed270, as depicted in FIG. 7, and processed by transaction server 240,depicted in FIG. 2. Market data may also be stored contemporaneously inMarket Data database 263.

In the preferred embodiment depicted in FIG. 6, traders may make aninvestment by selecting Trade button 504. Historical returns and timeseries data, from Market Data database 263 may be viewed by selectingDisplay button 505. Analytical tools for calculating opening orindicative returns or simulating market events are available by requestfrom Software Application Server 210 via ORB 230 and Transaction Server240 (depicted in FIG. 2) by selecting Analyze button 506 in FIG. 6. Asreturns change throughout the trading period, a trader may want todisplay how these returns have changed. As depicted in FIG. 6, theseintraday or intraperiod returns are available from Market Returnsdatabase 262 by selecting Intraday Returns button 507. In addition,marginal intra-period returns, as discussed previously, can be displayedusing the same data in Market Returns database 262 (FIG. 2). In apreferred embodiment, it is also possible for each trader to viewfinalized returns from Market Returns database 262.

In preferred embodiments that are not depicted, display 500 alsoincludes information identifying the group of contingent claims (such asthe claim type and event) available from the Contingent Claims Terms andConditions database 267 or current returns available from Market Returnsdatabase 262 (FIG. 2). In other preferred embodiments (e.g., anyembodiments of the present invention), display 500 includes means forrequesting other services which may be of interest to the trader, suchas the calculation of marginal returns, for example by selectingIntraday Returns button 507, or the viewing of historical data, forexample by selecting Historical Data button 505.

FIG. 7 depicts a preferred embodiment of the Market Data Feed 270 ofFIG. 2 in greater detail. In a preferred embodiment depicted in FIG. 7,which can be applied to other embodiments of the present invention,real-time data feed 600 comprises quotes of prices, yields, intradaytick graphs, and relevant market news and example sources. Historicaldata feed 610, which is used to supply market data database 263 withhistorical data, illustrates example sources for market time seriesdata, derived returns calculations from options pricing data, andinsurance claim data. Corporate action data feed 620 depicted in FIG. 7illustrates the types of discrete corporate-related data (e.g., earningsannouncements, credit downgrades) and their example sources which canform the basis for trading in groups of DBAR contingent claims of thepresent invention. In preferred embodiments, functions listed in process630 are implemented on transaction server 240 (FIG. 2) which takesinformation from data feeds 600, 610, and 620 for the purposes ofallocating returns, simulating outcomes, calculating risk, anddetermining event outcomes (as well as for the purpose of determininginvestment amounts).

FIG. 8 depicts a preferred embodiment of an illustrative graph ofimplied liquidity effects of investments in a group of DBAR contingentclaims. As discussed above, in preferred embodiments of the presentinvention, liquidity variations within a group of DBAR contingent claimimpose few if any costs on traders since only the finalized or closingreturns for a trading period matter to a trader's return. This contrastswith traditional financial markets, in which local liquidity variationsmay result in execution of trades at prices that do not fairly representfair fundamental value, and may therefore impose permanent costs ontraders.

Liquidity effects from investments in groups of DBAR contingent claims,as illustrated in FIG. 8, include those that occur when an investmentmaterially and permanently affects the distribution of returns acrossthe states. Returns would be materially and perhaps permanently affectedby a trader's investment if, for example, very close to the tradingperiod end time, a trader invested an amount in a state that representeda substantial percentage of aggregate amount previously invested in thatstate. The curves depicted FIG. 8 show in preferred embodiments themaximum effect a trader's investment can have on the distribution ofreturns to the various states in the group of DBAR contingent claims.

As depicted in FIG. 8, the horizontal axis, p, is the amount of thetrader's investment expressed as a percentage of the total amountpreviously invested in the state (the trade could be a multi-stateinvestment, but a single state is assumed in this illustration). Therange of values on the horizontal axis depicted in FIG. 8 has a minimumof 0 (no amount invested) to 10% of the total amount invested in aparticular state. For example, assuming the total amount invested in agiven state is $100 million, the horizontal axis of FIG. 8 ranges from anew investment amount of 0 to $10 million.

The vertical axis of FIG. 8 represents the ratio of the impliedbid-offer spread to the implied probability of the state in which a newinvestment is to be made. In a preferred embodiment, the impliedbid-offer spread is computed as the difference between the implied“offer” demand response, q_(i) ^(O)(ΔT_(i)), and the implied “bid”demand response, q_(i) ^(B)(ΔT_(i)), as defined above. In other words,values along the vertical axis depicted in FIG. 8 are defined by thefollowing ratio:

$\frac{{q_{i}^{O}\left( {\Delta \; T_{i}} \right)} - {q_{i}^{B}\left( {\Delta \; T_{i}} \right)}}{q_{i}}$

As displayed in FIG. 8, this ratio is computed using three differentlevels of q_(i), and the three corresponding lines for each level aredrawn over the range of values of p: the ratio is computed assuming alow implied q_(i)(q_(i)=0.091, denoted by the line marked S(p,l)), amiddle-valued q_(i)(q_(i)=0.333, denoted by the line marked S(p,m)), anda high value for q_(i)(q_(i)=0.833 denoted by the line marked S(p,h)),as shown.

If a trader makes an investment in a group of DBAR contingent claims ofthe present invention and there is not enough time remaining in thetrading period for returns to adjust to a fair value, then FIG. 8provides a graphical depiction, in terms of the percentage of theimplied state probability, of the maximum effect a trader's owninvestment can have on the distribution of implied state probabilities.The three separate curves drawn correspond to a high demand and highimplied probability (S(p,h)), medium demand and medium impliedprobability (S(p,m)), and low demand and low implied probability(S(p,l)). As used in this context, the term “demand” means the amountpreviously invested in the particular state.

The graph depicted in FIG. 8 illustrates that the degree to which theamount of a trader's investment affects the existing distribution ofimplied probabilities (and hence returns) varies with the amount ofdemand for the existing state as well as the amount of the trader'sinvestment. If the distribution of implied probabilities is greatlyaffected, this corresponds to a larger implied bid-offer spread, asgraphed on the vertical axis of the graph of FIG. 8. For example, forany given investment amount p, expressed as a percentage of the existingdemand for a particular state, the effect of the new investment amountis largest when existing state demand is smallest (line S(p,l),corresponding to a low demand/low implied probability state). Bycontrast, the effect of the amount of the new investment is smallestwhen the existing state demand is greatest (S(p,h), corresponding to ahigh demand/high implied probability state). FIG. 8 also confirms that,in preferred embodiments, for all levels of existing state demand, theeffect of the amount invested on the existing distribution of impliedprobabilities increases as the amount to be invested increases.

FIG. 8 also illustrates two liquidity-related aspects of groups of DBARcontingent claims of the present invention. First, in contrast to thetraditional markets, in preferred embodiments of the present inventionthe effect of a trader's investment on the existing market can bemathematically determined and calculated and displayed to all traders.Second, as indicated by FIG. 8, the magnitude of such effects are quitereasonable. For example, in preferred embodiments as depicted by FIG. 8,over a wide range of investment amounts ranging up to several percent ofthe existing demand for a given state, the effects on the market of suchinvestments amounts are relatively small. If the market has time toadjust after such investments are added to demand for a state, theeffects on the market will be only transitory and there may be no effecton the implied distribution of probabilities owing to the trader'sinvestment. FIG. 8 illustrates a “worst case” scenario by implicitlyassuming that the market does not adjust after the investment is addedto the demand for the state.

FIGS. 9 a to 9 c illustrate, for a preferred embodiment of a group ofDBAR contingent claims, the trader and credit relationships and howcredit risk can be quantified, for example in process 413 of FIG. 5.FIG. 9 a depicts a counterparty relationship for a traditional swaptransaction, in which two counterparties have previously entered into a10-year swap which pays a semi-annual fixed swap rate of 7.50%. Thereceiving counterparty 701 of the swap transaction receives the fixedrate and pays a floating rate, while the paying counterparty 702 paysthe fixed rate and receives the floating rate. Assuming a $100 millionswap trade and a current market fixed swap rate of 7.40%, based uponwell-known swap valuation principles implemented in software packagessuch as are available from Sungard Data Systems, the receivingcounterparty 701 would receive a profit of $700,000 while the payingswap counterparty 702 would have a loss of $700,000. The receiving swapcounterparty 701 therefore has a credit risk exposure to the paying swapcounterparty 702 as a function of $700,000, because the arrangementdepends on the paying swap party 702 meeting its obligation.

FIG. 9 b depicts illustrative trader relationships in which a preferredembodiment of a group of the DBAR contingent claims and exchangeeffects, as a practical matter, relationships among all the traders. Asdepicted in FIG. 9 b, traders C1, C2, C3, C4, and C5 each have investedin one or more states of a group of DBAR contingent claims, with definedstates S1 to S8 respectively corresponding to ranges of possibleoutcomes for the year swap rate, one year forward. In this illustration,each of the traders has a credit risk exposure to all the others inrelation to the amount of each trader's investment, how much of eachinvestment is on margin, the probability of success of each investmentat any point in time, the credit quality of each trader, and thecorrelation between and among the credit ratings of the traders. Thisinformation is readily available in preferred embodiments of DBARcontingent claim exchanges, for example in Trader and Account database261 depicted in FIG. 2, and can be displayed to traders in a formsimilar to tabulation 720 shown in FIG. 9 c, where the amount ofinvestment margin in each state is displayed for each trader, juxtaposedwith that trader's credit rating. For example, as depicted in FIG. 9 c,trader C1 who has a AAA credit rating has invested $50,000 on margin instate 7 and $100,000 on margin in state 8. In a preferred embodiment,the amount of credit risk borne by each trader can be ascertained, forexample using data from Market Data database 263 on the probability ofchanges in credit ratings (including probability of default), amountsrecoverable in case of default, correlations of credit rating changesamong the traders and the information displayed in tabulation 720.

To illustrate such determinations in the context of a group of DBARcontingent claims depicted in FIG. 9 c, the following assumptions aremade: (i) all the traders C1, C2, C3, C4 and C5 investing in the groupof contingent claims have a credit rating correlation of 0.9; (ii) theprobabilities of total default for the traders C1 to C5 are (0.001,0.003, 0.007, 0.01, 0.02) respectively; (iii) the implied probabilitiesof states 51 to S8 (depicted in FIG. 9 c) are(0.075,0.05,0.1,0.25,0.2,0.15,0.075,0.1), respectively. A calculationcan be made with these assumptions which approximates the total creditrisk for all of the traders in the group of the DBAR contingent claimsof FIG. 9 c, following Steps (i)-(vi) previously described for using VARmethodology to determine Credit-Capital-at-Risk.

Step (i) involves obtaining for each trader the amount of margin used tomake each trade. For this illustration, these data are assumed and aredisplayed in FIG. 9 c, and in a preferred embodiment, are available fromTrader and Account database 261 and Trade Blotter database 266.

Step (ii) involves obtaining data related to the probability of defaultand the percentage of outstanding margin loans that are recoverable inthe event of default. In preferred embodiments, this information isavailable from such sources as the JP Morgan CreditMetrics database. Forthis illustration a recovery percentage of zero is assumed for eachtrader, so that if a trader defaults, no amount of the margin loan isrecoverable.

Step (iii) involves scaling the standard deviation of returns (in unitsof the amounts invested) by the percentage of margin used for eachinvestment, the probability of default for each trader, and thepercentage not recoverable in the event of default. For thisillustration, these steps involve computing the standard deviations ofunit returns for each state, multiplying by the margin percentage ineach state, and then multiplying this result by the probability ofdefault for each trader. In this illustration, using the assumed impliedprobabilities for states 1 through 8, the standard deviations of unitreturns are: (3.5118, 4.359,3,1.732,2,2.3805,3.5118,3). In thisillustration these unit returns are then scaled by multiplying each by(a) the amount of investment on margin in each state for each trader,and (b) the probability of default for each trader, yielding thefollowing table:

S1 S2 S3 S4 S5 S6 S7 S8 C1, 175.59 300 AAA C2, AA 285.66 263.385 C3, AA1400 999.81 C4, A+ 2598 2000 C5, A 7023.6 4359 4800

Step (iv) involves using the scaled amounts, as shown in the above tableand a correlation matrix C_(s) containing a correlation of returnsbetween each pair of defined states, in order to compute aCredit-Capital-At-Risk. As previously discussed, this Step (iv) isperformed by first arranging the scaled amounts for each trader for eachstate into a vector U as previously defined, which has dimension equalto the number of states (e.g., 8 in this example). For each trader, thecorrelation matrix C_(s) is pre-multiplied by the transpose of U andpost-multiplied by U. The square root of the result is acorrelation-adjusted CCAR value for each trader, which represents theamount of credit risk contributed by each trader. To perform thesecalculations in this illustration, the matrix C_(s) having 8 rows and 8columns and 1's along the diagonal is constructed using the methodspreviously described:

$C_{s} = \begin{matrix}1 & {- {.065}} & {- {.095}} & {- {.164}} & {- {.142}} & {- {.12}} & {- {.081}} & {- {.095}} \\{- {.065}} & 1 & {- {.076}} & {- {.132}} & {- {.115}} & {- {.096}} & {- {.065}} & {- {.076}} \\{- {.095}} & {- {.076}} & 1 & {- {.192}} & {- {.167}} & {- {.14}} & {- {.095}} & {- {.111}} \\{- {.164}} & {- {.132}} & {- {.192}} & 1 & {- {.289}} & {- {.243}} & {- {.164}} & {- {.192}} \\{- {.142}} & {- {.115}} & {- {.167}} & {- {.289}} & 1 & {- {.21}} & {- {.142}} & {- {.167}} \\{- {.12}} & {- {.096}} & {- {.14}} & {- {.243}} & {- {.21}} & 1 & {- {.12}} & {- {.14}} \\{- {.081}} & {- {.065}} & {- {.095}} & {- {.164}} & {- {.142}} & {- {.12}} & 1 & {- {.095}} \\{- {.095}} & {- {.076}} & {- {.111}} & {- {.192}} & {- {.167}} & {- {.14}} & {- {.095}} & 1\end{matrix}$

The vectors U₁, U₂, U₃, U₄, and U₅ for each of the 5 traders in thisillustration, respectively, are as follows:

${U_{1} = {{\begin{matrix}0 \\0 \\0 \\0 \\0 \\0 \\175.59 \\300\end{matrix}\mspace{14mu} U_{2}} = {{\begin{matrix}0 \\0 \\0 \\0 \\0 \\285.66 \\263.385 \\0\end{matrix}\mspace{14mu} U_{3}} = {{\begin{matrix}0 \\0 \\0 \\0 \\1400 \\999.81 \\0 \\0\end{matrix}\mspace{14mu} U_{4}} = \begin{matrix}0 \\0 \\0 \\2598 \\2000 \\0 \\0 \\0\end{matrix}}}}}\mspace{14mu}$ $U_{5} = \begin{matrix}7023.6 \\4359 \\4800 \\0 \\0 \\0 \\0 \\0\end{matrix}$

Continuing with the methodology of Step (iv) for this illustration, fivematrix computations are performed as follows:

CCAR_(i)=√{square root over (U _(i) ^(T) *C _(s) *U _(i))}

for i=1 . . . 5. The left hand side of the above equation is the creditcapital at risk corresponding to each of the five traders.

Pursuant to Step (v) of the CCAR methodology as applied to this example,the five CCAR values are arranged into a column vector of dimensionfive, as follows:

$w_{CCAR} = \begin{matrix}332.9 \\364.58 \\1540.04 \\2783.22 \\8820.77\end{matrix}$

Continuing with this step, a correlation matrix (CCAR) with a number ofrows and columns equal to the number of traders is constructed whichcontains the statistical correlation of changes in credit ratingsbetween every pair of traders on the off-diagonals and 1's along thediagonal. For the present example, the final Step (vi) involves thepre-multiplication of CCAR by the transpose of W_(CCAR) and the postmultiplication of C_(CCAR) by w_(CCAR), and taking the square root ofthat product, as follows:

CCAR_(TOTAL)=√{square root over (C _(CCAR) ^(T) *C _(CCAR) ^(T) *w_(CCAR))}

In this illustration, the result of this calculation is:

${CCAR}_{TOTAL} = {\sqrt{\begin{matrix}332.9 & 364.58 & 1540.04 & 2783.22 & {8820.77*\begin{matrix}1 & {.9} & {.9} & {.9} & {.9} & 332.9 \\{.9} & 1 & {.9} & {.9} & {.9} & 364.58 \\{.9} & {.9} & 1 & {.9} & {{.9}*} & 1540.04 \\{.9} & {.9} & {.9} & 1 & {.9} & 2783.22 \\{.9} & {.9} & {.9} & {.9} & 1 & 8820.77\end{matrix}}\end{matrix}} = 1342.74}$

In other words, in this illustration, the margin total and distributionshowing in FIG. 9 c has a single standard deviationCredit-Capital-At-Risk of $13,462.74. As described previously in thediscussion of Credit-Capital-At-Risk using VAR methodology, this amountmay be multiplied by a number derived using methods known to those ofskill in the art in order to obtain a predetermined percentile of creditloss which a trader could believe would not be exceeded with apredetermined level of statistical confidence. For example, in thisillustration, if a trader is interested in knowing, with a 95%statistical confidence, what loss amount would not be exceeded, thesingle deviation Credit-Capital-At-Risk figure of $13,462.74 would bemultiplied by 1.645, to yield a figure of $22,146.21.

A trader may also be interested in knowing how much credit risk theother traders represent among themselves. In a preferred embodiment, thepreceding steps (i)-(vi) can be performed excluding one or more of thetraders. For example, in this illustration, the most risky trader,measured by the amount of CCAR associated with it, is trader C5. Theamount of credit risk due to C1 through C4 can be determined byperforming the matrix calculation of Step (v) above, by entering 0 forthe CCAR amount of trader C5. This yields, for example, a CCAR fortraders C1 through C4 of $4,870.65.

FIG. 10 depicts a preferred embodiment of a feedback process forimproving of a system or exchange for implementing the present inventionwhich can be used with other embodiments of the present invention. Asdepicted in FIG. 10, in a preferred embodiment, closing and intraperiodreturns from Market Returns database 262 and market data from MarketData database 263 (depicted in FIG. 2) are used by process 910 for thepurpose of evaluating the efficiency and fairness of the DBAR exchange.One preferred measure of efficiency is whether a distribution of actualoutcomes corresponds to the distribution as reflected in the finalizedreturns. Distribution testing routines, such as Kolmogorov-Smirnofftests, preferably are performed in process 910 to determine whether thedistributions implied by trading activity in the form of returns acrossthe defined states for a group of DBAR contingent claims aresignificantly different from the actual distributions of outcomes forthe underlying events, experienced over time. Additionally, in preferredembodiments, marginal returns are also analyzed in process 910 in orderto determine whether traders who make investments late in the tradingperiod earn returns statistically different from other traders. These“late traders,” for example, might be capturing informational advantagesnot available to early traders. In response to findings from analyses inprocess 910, a system according to the present invention for trading andinvesting in groups of the DBAR contingent claims can be modified toimprove its efficiency and fairness. For example, if “late traders” earnunusually large profits, it could mean that such a system is beingunfairly manipulated, perhaps in conjunction with trading in traditionalsecurity markets. Process 920 depicted in FIG. 10 represents a preferredembodiment of a counter-measure which randomizes the exact time at whicha trading period ends for the purposes of preventing manipulation ofclosing returns. For example, in a preferred embodiment, an exchangeannounces a trading closing end time falling randomly between 2:00 p.m.and 4:00 p.m. on a given date.

As depicted in FIG. 10, process 923 is a preferred embodiment of anotherprocess to reduce risk of market manipulation. Process 923 representsthe step of changing the observation period or time for the outcome. Forexample, rather than observing the outcome at a discrete time, theexchange may specify that a range of times for observation will used,perhaps spanning many hours, day, or weeks (or any arbitrary timeframe), and then using the average of the observed outcomes to determinethe occurrence of a state.

As further depicted in FIG. 10, in response to process 910, steps couldbe taken in process 924 to modify DRFs in order, for example, toencourage traders to invest earlier in a trading period. For example, aDRF could be modified to provide somewhat increased returns to these“early” traders and proportionately decreased returns to “late” traders.Similarly for digital options, an OPF could be modified to providesomewhat discounted prices for “early” traders and proportionatelymarked-up prices for “late” traders. Such incentives, and othersapparent to those skilled in the art, could be reflected in moresophisticated DRFs.

In a preferred embodiment depicted in FIG. 10, process 921 represents,responsive to process 910, steps to change the assumptions under whichopening returns are computed for the purpose of providing better openingreturns at the opening of the trading period. For example, the resultsof process 910 might indicate that traders have excessively traded theextremes of a distribution in relation to actual outcomes. There isnothing inherently problematic about this, since trader expectations forfuture possible outcomes might reflect risk preferences that cannot beextracted or analyzed with actual data. However, as apparent to one ofskill in the art, it is possible to adjust the initial returns toprovide better estimates of the future distribution of states, by, forexample, adjusting the skew, kurtosis, or other statistical moments ofthe distribution.

As depicted in FIG. 10, process 922 illustrates changing entirely thestructure of one or more groups of DBAR contingent claims. Such acountermeasure can be used on an ad hoc basis in response to graveinefficiencies or unfair market manipulation. For example, process 922can include changes in the number of trading periods, the timing oftrading periods, the duration of a group of DBAR contingent claims, thenumber of and nature of the defined state partitions in order to achievebetter liquidity and less unfair market manipulation for groups of DBARcontingent claims of the present invention.

As discussed above (Section 6), in a preferred embodiment of a DBARDigital Options Exchange (“DBAR DOE”), traders may buy and “sell”digital options, spreads, and strips by either placing market orders orlimit orders. A market order typically is an order that isunconditional, i.e., it is executed and is viable regardless of DBARcontingent claim “prices” or implied probabilities. A limit order, bycontrast, typically is a conditional investment in a DBAR DOE in whichthe trader specifies a condition upon which the viability or execution(i.e., finality) of the order depends. In a preferred embodiment, suchconditions typically stipulate that an order is conditional upon the“price” for a given contingent claim after the trading period has beencompleted upon fulfillment of the trading period termination criteria.At this point, all of the orders are processed and a distribution ofDBAR contingent claim “prices”—which for DBAR digital options is theimplied probability that the option is “in the money”—are determined.

In a preferred embodiment of a DBAR DOE of the present invention, limitorders may be the only order type that is processed. In a preferredembodiment, limit orders are executed and are part of the equilibriumfor a group of DBAR contingent claims if their stipulated “price”conditions (i.e., probability of being in the money) are satisfied. Forexample, a trader may have placed limit buy order at 0.42 for MSFTdigital call options with a strike price of 50. With a the limitcondition at 0.42, the trader's order will be filled only if the finalDBAR contingent claim distribution results in the 50 calls having a“price” which is 0.42 or “better,” which, for a buyer of the call, means0.42 or lower.

Whether a limit order is included in the final DBAR equilibrium affectsthe final probability distribution or “prices.” Since those “prices”determine whether such limit orders are to be executed and thereforeincluded in the final equilibrium, in a preferred embodiment aniterative procedure, as described in detail below, may be carried outuntil an equilibrium is achieved.

As described above, in a preferred embodiment, A DBAR DOE equilibriumresults for a contract, or group of DBAR contingent claims includinglimit orders, when at least the following conditions have been met:

-   -   (1) At least some buy (“sell”) orders with a limit “price”        greater (less) than or equal to the equilibrium “price” for the        given option, spread or strip are executed or “filled.”    -   (2) No buy (“sell”) orders with limit “prices” less (greater)        than the equilibrium “price” for the given option, spread or        strip are executed.    -   (3) The total amount of executed lots equals the total amount        invested across the distribution of defined states.    -   (4) The ratio of payouts should each constituent state of a        given option, spread, or strike occur is as specified by the        trader, (including equal payouts in the case of digital        options), within a tolerable degree of deviation.    -   (5) Conversion of filled limit orders to market orders for the        respective filled quantities and recalculating the equilibrium        does not materially change the equilibrium.    -   (6) Adding one or more lots to any of the filled limit orders        converted to market orders in step (5) and recalculating of the        equilibrium “prices” results in “prices” which violate the limit        “price” of the order to which the lot was added (i.e., no more        lots can be “squeaked in” without forcing market prices to go        above the limit “prices” of buy orders or below the limit        “prices” of sell orders).

In a preferred embodiment, the DBAR DOE equilibrium is computed throughthe application of limit and market order processing steps, multistatecomposite equilibrium calculation steps, steps which convert “sell”orders so that they may be processed as buy orders, and steps whichprovide for the accurate processing of limit orders in the presence oftransaction costs. The descriptions of FIGS. 11-18 which follow explainthese steps in detail. Generally speaking, in a preferred embodiment, asdescribed in Section 6, the DBAR DOE equilibrium including limit ordersis arrived at by:

-   -   (i) converting any “sell” orders to buy orders;    -   (ii) aggregating the buy orders (including the converted “sell”        orders) into groups for which the contingent claims specified in        the orders share the same range of defined states;    -   (iii) adjusting the limit orders for the effect of transaction        costs by subtracting the order fee from the order's limit        “price;”    -   (iv) sorting the orders upon the basis of the (adjusted) limit        order “prices” from best (highest) to worst (lowest);    -   (v) searching for an order with a limit “price” better (i.e.,        higher) than the market or current equilibrium “price” for the        contingent claim specified in the order;    -   (vi) if such a better order can be found, adding as many        incremental value units or “lots” of that order for inclusion        into the equilibrium calculation as possible without newly        calculated market or equilibrium “price” exceeding the specified        limit “price” of the order (this is known as the “add” step);    -   (vii) searching for an order with previously included lots which        now has a limit “price” worse than the market “price” for the        contingent claim specified in the order (i.e., lower than the        market “price”);    -   (viii) removing the smallest number of lots from the order with        the worse limit “price” so that the newly calculated equilibrium        “price,” after such iterative removal of lots, is just below the        order's limit “price” (this is known as the “prune” step, in the        sense that lots previously added are removed or “pruned” away);    -   (ix) repeating the “add” and “prune” steps until no further        orders remain which are either better than the market which have        lots to add, or worse than the market which have lots to remove;    -   (x) taking the “prices” resulting from the final equilibrium        resulting from step (ix) and adding any applicable transaction        fee to obtain the offer “price” for each respective contingent        claim ordered and subtracting any applicable transaction fee to        obtain the bid “price” for each respective contingent claim        ordered; and    -   (xi) upon fulfillment of all of the termination criteria related        to the event of economic significance or state of a selected        financial product, allocating payouts to those orders which have        investments on the realized state, where such payouts are        responsive to the final equilibrium “prices” of the orders'        contingent claims and the transaction fees for such orders.

Referring to FIG. 11, illustrative data structures are depicted whichmay be used in a preferred embodiment to store and manipulate the datarelevant to the DBAR DOE embodiment and other embodiments of the presentinvention. The data structure for a “contract” or group of DBARcontingent claims, shown in 1101, contains data members which store datawhich are relevant to the construction of the DBAR DOE contract or groupof claims. Specifically, the contract data structure contains (i) thenumber of defined states (contract.numStates); (ii) the total amountinvested in the contract at any given time (contract.totallnvested);(iii) the aggregate profile trade investments required to satisfy theaggregate profile trade requests for profile trades (a type of tradewhich is described in detail below) (iv) the aggregate payout requestsmade by profile trades; (v) the total amount invested or allocated ineach defined state at any given time (contract.stateTotal); (vi) thenumber of orders submitted at any given time (contract.numOrders); and(vii) a list of the orders, which is itself a structure containing datarelevant to the orders (contract.orders[ ]).

A preferred embodiment of “order” data structures, shown in 1102 of FIG.11, illustrates the data which are typically needed to process atrader's order using the methods of the DBAR DOE of the presentinvention. Specifically, the order data structure contains the followingrelevant members for order processing:

-   -   (i) the amount of the order which the trader desires to        transact. For orders which request the purchase (“buys”) of a        digital option, strip, or spread, the amount is interpreted as        the amount to invest in the desired contingent claim. Thus, for        buys, the order amount is analogous to the option premium for        conventional options. For orders which request “sales” of a DBAR        contingent claim, the order amount is to be interpreted as the        amount of net payout that the trader desires to “sell.” Selling        a net payout in the context of a DBAR DOE of the present        invention means that the loss that a trader suffers should the        digital option, strip or spread “sold” expire in the money is        equal to the payout “sold.” In other words, by selling a net        payout, the trader is able to specify the amount of net loss        that would occur should the option “sold” expire in the money.        If the contingent claim “sold” expires out of the money, the        trader would receive a profit equal to the net payout multiplied        by the ratio of (a) the final implied probability of the option        expiring in the money and (b) the implied probability of the        option expiring out of the money. In other words, in a preferred        embodiment of a DBAR DOE, “buys are for premium, and sells are        for net payout” which means that buy orders and sell orders in        terms of the order amount are interpreted somewhat differently.        For a buy order, the premium is specified and the payout, should        the option expire in the money, is not known until all of the        predetermined termination criteria have been met at the end of        trading. For a “sell” order, in contrast, the payout to be        “sold” is specified (and is equal to the net loss should the        option “sold” expire in the money), while the premium, which is        equal to the trader's profit should the option “sold” expire out        of the money, is not known until all of the predetermined        termination criteria have been met (e.g., at the end of        trading);    -   (ii) the amount which must be invested in each defined state to        generate the desired digital option, spread or strip specified        in the order is contained in data member order.invest[ ];    -   (iii) the data members order.buySell indicates whether the order        is a buy or a “sell”;    -   (iv) the data members order.marketLimit indicates whether the        order is a limit order whose viability for execution is        conditional upon the final equilibrium “price” after all        predetermined termination criteria have been met, or a market        order, which is unconditional;    -   (v) the current equilibrium “price” of the digital option,        spread or strip specified in the order;    -   (vi) a vector which specifies the type of contingent claim to be        traded (order.ratio[ ]). For example, in a preferred embodiment        involving a contract with seven defined states, an order for a        digital call option which would expire in the money should any        of the last four states occur would be rendered in the data        member order. ratio[ ] as order.ratio[0,0,0,1,1,1,1] where the        1's indicate that the same payout should be generated by the        multistate allocation process when the digital option is in the        money, and the 0's indicate that the option is out of the money,        or expires on one of the respective out of the money states. As        another example in a preferred embodiment, a spread which is in        the money should states either states 1,2, 6, or 7 occur would        be rendered as order.ratio[1,1,0,0,0,1,1]. As another example in        a preferred embodiment, a digital option strip, which allows a        trader to specify the relative ratios of the final payouts owing        to an investment in such a contingent claim would be rendered        using the ratios over which the strip is in the money. For        example, if a trader desires a strip which pays out three times        much as state 3 should state 1 occur, and twice as much as state        3 if state 2 occurs, the strip would be rendered as        order.ratio[3,2,1,0,0,0,0];    -   (vii) the amount of the order than can be executed or filled at        equilibrium. For market orders, the entire order amount will be        filled, since such orders are unconditional. For limit orders,        none, all, or part of the order amount may be filled depending        upon the equilibrium “prices” prevailing when the termination        criteria are fulfilled;    -   (viii) the transaction fee applicable to the order;    -   (ix) the payout for the order, net of fees, after all        predetermined termination criteria have been met; and    -   (x) a data structure which, for trades of the profile type        (described below in detail), contains the desired amount of        payout requested by the order should each state occur.

FIG. 12 depicts a logical diagram of the basic steps for limit andmarket order processing in a preferred embodiment of a DBAR DOE of thepresent invention. Step 1201 of FIG. 12 loads the relevant data into thecontract and order data structures of FIG. 11. Step 1202 initializes theset of DBAR contingent claims, or the “contract,” by placing initialamounts of value units (i.e., initial liquidity) in each state of theset of defined states. The placement of initial liquidity avoids asingularity in any of the defined states (e.g., an invested amount in agiven defined state equal to zero) which may tend to impede multistateallocation calculations. The initialization of step 1202 may be done ina variety of different ways. In a preferred embodiment, a small quantityof value units is placed in each of the defined states. For example, asingle value unit (“lot”) may be placed in each defined state where thesingle value unit is expected to be small in relation to the totalamount of volume to be transacted. In step 1202 of FIG. 12, the initialvalue units are represented in the vector init[contract.numStates].

In a preferred embodiment, step 1203 of FIG. 12 invokes the functionconvertSales( ) which converts all of the “sell” orders to complementarybuy orders. The function convertSales( ) is described in detail in FIG.15, below. After the completion of step 1203, all of the orders forcontingent claims—whether buy or “sell” orders, can be processed as buyorders.

In a preferred embodiment, step 1204 groups these buy orders based uponthe distinct ranges of states spanned by the contingent claims specifiedin the orders. The range of states comprising the order are contained inthe data member order.ratio[ ] of the order data structure 1102 depictedin FIG. 11.

In a preferred embodiment, for each order[j] there is associated avector of length equal to the number of defined states in the contractor group of DBAR contingent claims (contract.numStates). This vector,which is stored in order[j].ratio[ ], contains integers which indicatethe range of states in which an investment is to be made in order togenerate the expected payout profile of the contingent claim desired bythe trader placing the order.

In a preferred embodiment depicted in FIG. 12, a separate grouping instep 1204 is required for each distinct order[j].ratio[ ] vector. Twoorder[j].ratio[ ] vectors are distinct for different orders when theirdifference yields a vector that does not contain zero in every element.For example, for a contract which contains seven defined states, adigital put option which spans that first three states has anorder[1].ratio[ ] vector equal to (1,1,1,0,0,0,0). A digital call optionwhich spans the last five states has an order[2].ratio[ ] vector equalto (0,0,1,1,1,1,1). Because the difference of these two vectors is equalto (1,1,0,−1,−1,−1,−1), these two orders should be placed into distinctgroups, as indicated in step 1204.

In a preferred embodiment depicted in FIG. 12, step 1204 aggregatesorders into relevant groups for processing. For the purposes ofprocessing limit orders: (i) all orders may be treated as limit orderssince orders without limit “price” conditions, e.g., “market orders,”can be rendered as limit buy orders (including “sale” orders convertedto buy orders in step 1203) with limit “prices” of 1, and (ii) all ordersizes are processed by treating them as multiple orders of the smallestvalue unit or “lot.”

The relevant groups of step 1204 of FIG. 12 are termed “composite” sincethey may span, or comprise, more than one of the defined states. Forexample, the MSFT

Digital Option contract depicted above in Table 6.2.1, for example, hasdefined states (0,30], (30,40], (40,50], (50,60], (60, 70], (70, 80],and (80,00]. The 40 strike call options therefore span the five states(40,50], (50,60], (60, 70], (70, 80], and (80,00]. A “sale” of a 40strike put, for example, would be converted at step 1203 into acomplementary buy of a 40 strike call (with a limit “price” equal to oneminus the limit “price” of the sold put), so both the “sale” of the 40strike put and the buy of a 40 strike call would be aggregated into thesame group for the purposes of step 1204 of FIG. 12.

In the preferred embodiment depicted in FIG. 12, step 1205 invokes thefunction feeAdjustOrders( ) This function is required so as toincorporate the effect of transaction or exchange fees for limit orders.The function feeAdjustOrders( ) shown in FIG. 12, described in detailwith reference to FIG. 16, basically subtracts from the limit “price” ofeach order the fee for that order's contingent claim. The limit “price”is then set to this adjusted, lower limit “price” for the purposes ofthe ensuing equilibrium calculations.

At the point of step 1206 of the preferred embodiment depicted in FIG.12, all of the orders may be processed as buy orders (because any “sell”orders have been converted to buy orders in step 1203 of FIG. 12) andall limit “prices” have been adjusted (with the exception of marketorders which, in a preferred embodiment of the DBAR DOE of the presentinvention, have a limit “price” equal to one) to reflect transactioncosts equal to the fee specified for the order's contingent claim (ascontained in the data member order[j].fee). For example, consider thesteps depicted in FIG. 12 leading up to step 1206 on three hypotheticalorders: (1) a buy order for a digital call with strike price of 50 witha limit “price” of 0.42 for 100,000 value units (lots) (on theillustrative MSFT example described above); (2) a “sale” order for adigital put with a strike price of 40 with a limit price of 0.26 for200,000 value units (lots); and (3) a market buy order for a digitalspread which is in the money should MSFT stock expire greater than orequal to 40 and less than or equal to 70. In a preferred embodiment, therepresentations of the range of states for the contingent claimsspecified in the three orders are as follows: (1) buy order for50-strike digital call: order[1].ratio[ ]=(0,0,0,1,1,1,1); (2) “sell”order for 40-strike digital put: order[2].ratio[ ]=(0,0,1,1,1,1,1); and(3) market buy order for a digital spread in the money on the interval[40,70): order[3].ratio[ ]=(0,0,1,1,1,1,0). Also in this preferredembodiment, the “sell” order of the put covers the states as a“converted” buy order which are complementary to the states being sold(sold states=order.ratio[ ]=(1,1,0,0,0,0,0)), and the limit “price” ofthe converted order is equal to one minus the limit “price” of theoriginal order (i.e., 1−0.26=0.74). Then in a preferred embodiment, allof the orders' limit “prices” are adjusted for the effect of transactionfees so that, assuming a fee for all of the orders equal to 0.0005(i.e., 5 basis points of notional payout), the fee-adjusted limit pricesof the orders are equal to (1) for the 50-strike call: 0.4195(0.42-0.0005); (2) for the converted sale of 40-strike put: 0.7395(1-0.26-0.0005); and (3) for the market order for digital spread: 1(limit “price” is set to unity). In a preferred embodiment depicted inFIG. 12, step 1204 then would aggregate these hypothetical orders intodistinct groups, where orders in each group share the same range ofdefined states which comprise the orders' contingent claim. In otherwords, as a result of step 1204, each group contains orders which haveidentical vectors in order.ratio[ ]. For the illustrative threehypothetical orders, the orders would be placed as a result of step 1204into three separate groups, since each order ranges over distinct setsof defined states as indicated in their respective order[j].ratio[ ]vectors (i.e., (0,0,0,1,1,1,1), (0,0,1,1,1,1,1), and (0,0,1,1,1,1,0),respectively).

For the purposes of step 1206 of the preferred embodiment depicted inFIG. 12, all of the order have been converted to buy orders and have hadtheir limit “prices” adjusted to reflect transaction fees, if any. Inaddition, such orders have been placed into groups which share the samerange of defined states which comprise the contingent claim specified inthe orders (i.e., have the same order[j].ratio[ ] vector). In thispreferred embodiment depicted in FIG. 12, step 1206 sorts each group'sorders based upon their fee-adjusted limit “prices,” from best (highest“prices”) to worst (lowest “prices”). For example, consider a set oforders in which only digital calls and puts have been ordered, both tobuy and to “sell,” for the MSFT example of Table 6.2.1 in which strikeprices of 30, 40, 50, 60, 70, and 80 are available. A “sale” of a callis converted to a buy of a put, and a “sale” of a put is converted intoa purchase of a call by step 1204 of the preferred embodiment depictedin FIG. 12. Thus, in this embodiment all of the grouped orderspreferably are grouped in terms of calls and puts at the indicatedstrike prices of the orders.

The grouped orders, after conversion and adjustment for fees, can beillustrated in Diagram 1 shown in FIG. 27, which depicts the results ofa grouping process for a set of illustrative and assumed digital putsand calls.

Referring to Diagram 1 the call and put limit orders have been groupedby strike price (distinct order[j].ratio[ ] vectors) and then orderedfrom “best price” to “worst,” moving away from the horizontal axis. Asshown in the table, “best price” for buy orders are those with higherprices (i.e., buyers with a higher willingness to pay). Diagram 1includes “sales” of puts which have been converted to complementarypurchases of calls and “sales” of calls which have been converted tocomplementary purchases of puts, i.e., all orders for the purposes ofDiagram 1 may be treated as buy orders.

For example, as depicted in Diagram 1 the grouping which includes thepurchase of the 40 calls (labeled “C40”) would also include anyconverted “sales” of the 40 puts (i.e., “sale” of the 40 puts has anorder.ratio[ ] vector which originally is equal to (1,1,0,0,0,0,0) andis then converted to the complementary order.ratio[ ] vector(0,0,1,1,1,1,1) which corresponds to the purchase of a 40-strike call).

Diagram 1 illustrates the groupings that span distinct sets of definedstates with a vertical bar. The labels within each vertical bar inDiagram 1 such as “C50”, indicate whether the grouping corresponds to acall (“C”) or put (“P”) and the relevant strike price, e.g., “C50”indicates a digital call option with strike price of 50. The horizontallines within each vertical bar shown on Diagram 1 indicates the sortingby price within each group. Thus, for the vertical bar above thehorizontal axis marked “C50” in Diagram 1, there are five distinctrectangular groupings within the vertical bar. Each of these groupingsis an order for the digital call options with strike price 50 at aparticular limit “price.” By using the DBAR methods of the presentinvention, there is no matching of buyers and “sellers,” or buy ordersand “sell” orders, which is typically required in the traditionalmarkets in order for transactions to take place. For example, Diagram 1illustrates a set of orders that contains only buy orders for thedigital puts struck at 70 (“P70”).

In a preferred embodiment of a DBAR DOE of the present invention, theaggregation of orders into groups referred to by step 1204 of thepreferred embodiment depicted in FIG. 12 corresponds to DBAR digitaloptions, spread, and strip trades which span distinct ranges of thedefined states. For example, the 40 puts and the 40 calls arerepresented as distinct state sets since they span or comprise differentranges of defined states.

Proceeding with the next step of the preferred embodiment depicted inFIG. 12, step 1207 queries whether there is at least a single orderwhich has a limit “price” which is “better” than the current equilibrium“price” for the ordered option. In a preferred embodiment for the firstiteration of step 1207 for a trading period for a group of DBARcontingent claims, the current equilibrium “prices” reflect theplacement of the initial liquidity from step 1202. For example, with theseven defined states of the MSFT example described above, one value unitmay have been initialized in each of the seven defined states. The“prices” of the 30, 40, 50, 60, 70, and 80 digital call options, aretherefore 6/7, 5/7, 4/7, 3/7, 2/7, and 1/7, respectively. The initial“prices” of the 30, 40, 50, 60, 70, and 80 digital puts are 1/7, 2/7,3/7, 4/7, 5/7, 6/7, respectively. Thus, step 1207 may identify a buyorder for a 60 digital call option with limit “price” greater than 3/7(0.42857) or a “sell” order, for example, for the 40 digital put optionwith limit “price” less than 2/7 (0.28571) (which would be convertedinto a buy order of the 40 calls with limit “price” of 5/7 (i.e.,1-2/7)). In a preferred embodiment an order's limit “price” or impliedprobability would take into account transaction or exchange fees, sincethe limit “prices” of the original orders would have been alreadyadjusted by the amount of the transaction fee (as contained inorder[j].fee) from step 1205 of FIG. 12, where the function fee AdjustOrders( ) is invoked.

As discussed above, transaction or exchange fees, and consequentlybid/offer “prices” or implied probability, can be computed in a varietyof ways. In a preferred embodiment, such fees are computed as a fixedpercentage of the total amount invested over all of the defined states.The offer (bid) side of the market for a given digital option (or stripor spread) is computed in this embodiment by taking the total amountinvested less (plus) this fixed percentage, and dividing it by the totalamount invested over the range of states comprising the given option (orstrip or spread). This reciprocal of this quantity then equals the offer(bid) “price” in this embodiment. In another preferred embodiment,transaction fees are computed as a fixed percentage of the payout of agiven digital option, strip or spread. In this embodiment, if thetransaction fee is f basis points of the payout, then the offer (bid)price is equal to the total amount invested over the range of statecomprising the digital option (strip or spread) plus (minus) f basispoints. For example, assume that f is equal to 5 basis points or 0.0005.Thus, the offer “price” of an in-the-money option whose equilibrium“price” is 0.50 might be equal to 0.50+0.0005 or 0.5005 and the bid“price” equal to 0.50−0.0005 or 0.4995. An out-of-the-money optionhaving an equilibrium “price” equal to 0.05 might therefore have anoffer “price” equal to 0.05+0.0005 or 0.0505 and a bid “price” equal to0.05−0.0005 or 0.0495. Thus, the embodiment in which transaction feesare a fixed percentage of the payout yields bid/offer spreads that are ahigher percentage of the out-of-the-money option “prices” than of thein-the-money option prices.

The bid/offer “prices” affect not only the costs to the trader of usinga DBAR digital options exchange, but also the nature of the limit orderprocess. Buy limit orders (including those buy orders which areconverted “sells”) must be compared to the offer “prices” for theoption, strip or spread contained in the order. Thus a buy order has alimit “price” which is “better” than the market if the limit “price”condition is greater than or equal to the offer side of the market forthe option specified in the order. Conversely, a “sell” order has alimit “price” which is better than the market if the limit “price”condition is less than or equal to the bid side of the market for theoption specified in the order. In the preferred embodiment depicted inFIG. 12, the effect of transaction fees is captured by the adjustment ofthe limit “prices” in step 1205, in that in equilibrium an order shouldbe filled only if its limit “price” is better than the offer “price”,which includes the transaction fee.

In the preferred embodiment depicted in FIG. 12, if step 1207 identifiesat least one order which has a limit “price” better than the current setof equilibrium “prices” (whether the initial set of “prices” upon thefirst iteration or the “prices” resulting from subsequent iterations)then step 1208 invokes the function fillRemoveLots. The functionfillRemoveLots, when called with the first parameter equal to one as instep 1208, will attempt to add lots from the order identified in step1207 which has limit “price” better than the current set of equilibriumprices. The fillRemoveLots function is described in detail in FIG. 17,below. Basically, the function finds the number of lots of the orderthan can be added for a buy order (including all “sale” order convertedto buy orders) such that when a new equilibrium set of “prices” iscalculated for the group of DBAR contingent claims with the added lots(by invoking the function compEq( ) of FIG. 13), no further lots can beadded without causing the new equilibrium “price” with those added lotsto exceed the limit “price” of the buy order being filled.

In preferred embodiments, finding the maximum amount of lots to add sothat the limit “price” is just better than the new equilibrium isaccomplished using the method of binary search, as described in detailwith reference to FIG. 17, below. Also in preferred embodiments the stepof “filling” lots refers to the execution, incrementally anditeratively, using the method of binary search, of that part of theorder quantity that can be executed or “filled.” In a preferredembodiment, the filling of a buy order therefore requires the testing,via the method of binary search, to determine whether additional unitlots can be added over the relevant range of defined states spanning theparticular option for the purposes of equilibrium calculation, withoutcausing the resulting equilibrium “price” for the order to exceed thelimit “price.”

In the preferred embodiment depicted in FIG. 12, step 1209 is executedfollowing step 1208 if lots are filled, or following step 1207 if noorders were identified with limit “prices” which are better than thecurrent equilibrium “prices.” Step 1209 of FIG. 12 identifies ordersfilled at least partially at limit “prices” which are worse (i.e., less)than the current equilibrium “prices.” In preferred embodiments, thefilling of lots in step 1208, if performed prior to step 1209, involvesthe iterative recalculation of the equilibrium “prices” by invoking thefunction compEq( ) which is described in detail with reference to FIG.13.

In the preferred embodiment depicted in FIG. 12, the equilibriumcomputations in step 1208 performed in the process of filling lots maycause a change in the equilibrium “prices” which in turn may causepreviously filled orders to have limit “prices” which are now worse(i.e., lower) than the new equilibrium. Step 1209 identifies theseorders. In order for the order to comply with the equilibrium, its limit“price” may not be worse (i.e., less) than the current equilibrium.Thus, in a preferred embodiment of the DBAR

DOE of the present invention, lots for such an order are removed. Thisis performed in step 1210 with the invocation of functionfillRemoveLots. Similar to step 1208, in a preferred embodiment theprocessing step 1210 uses the method of binary search to find theminimum amount of lots to be removed from the quantity of the order thathas already been filled such that the order's limit “price” is no longerworse (i.e., less) than the equilibrium “price,” which is recomputediteratively. For buy orders and all buy orders converted from “sell”orders processed in step 1210, a new filled quantity is found which issmaller than the original filled quantity so that the buy order's newequilibrium “price” does not exceed the buy order's specified limit“price.”

The logic of steps 1207-1210 of FIG. 12 may be summarized as follows. Anorder is identified which can be filled (step 1207), i.e., an orderwhich has a limit “price” better than the current equilibrium “price”for the option specified in the order. If such an order is identified,it is filled to the maximum extent possible without violating the limit“price” condition of the order itself (step 1208). A buy order's limit“price” condition is violated if an incremental lot is filled whichcauses the equilibrium “price,” taking account of this additional lot,to exceed the buy order's limit “price.” Any previously filled ordersmay now have limit order conditions that are violated as a result oflots being filled in step 1208. These orders are identified, one orderat a time, in step 1209. The filled amounts of such orders with violatedlimit order “price” conditions are reduced or “pruned” so that the limitorder “price” conditions are no longer violated. This “pruning” isperformed in step 1210. The steps 1207 to 1210 constitute an “add andprune” cycle in which an order with a limit “price” better than theequilibrium of the current iteration has its filled amount increased,followed by the reduction or pruning of any filled amounts for orderswith a limit “price” condition which is worse than the equilibrium“price” of the current iteration. In preferred embodiment, the “add andprune” cycle continues until there remain no further orders with limit“price” conditions which are either better or worse than theequilibrium, i.e., no further adding or pruning can be performed.

When no further adding or pruning can be performed, an equilibrium hasbeen achieved, i.e., all of the orders with limit “prices” worse thanthe equilibrium are not executed and at least some part of all of theorders with limit “prices” better or equal to the equilibrium areexecuted. In the preferred embodiment of FIG. 12, completion of the “addand prune” cycle terminates limit and market order processing asindicated in step 1211. The final “prices” of the equilibriumcalculation resulting from the “add and prune” cycle of steps 1207-1210can be designated as the mid-market “prices.” The bid “prices” for eachcontingent claim are computed by subtracting a fee from the mid-market“prices,” and the offer “prices” are computed by adding a fee to themid-market “prices.” Thus, equilibrium mid-market, bid, and offer“prices” may then be published to traders in a preferred embodiment of aDBAR DOE.

Referring now to the preferred embodiment of a method of compositemultistate equilibrium calculation depicted FIG. 13, the functioncompEq( ) which is a multistate allocation algorithm, is described. In apreferred embodiment of a DBAR DOE, digital options span or comprisemore than one defined state, with each of the defined statescorresponding to at least one possible outcome of an event of economicsignificance or a financial instrument. As depicted in Table 6.2.1above, for example, the MSFT digital call option with strike price of 40spans the five states above 40 or (40,50], (50,60], (60, 70], (70, 80],and (80,00]. To achieve a profit and loss scenario that tradersconventionally expect from a digital option, in a preferred embodimentof the present invention a digital option investment of value unitsdesignates a set of defined states and a desired return-on-investmentfrom the designated set of defined states, and the allocation ofinvestments across these states is responsive to the desiredreturn-on-investment from the designated set of defined states. For adigital option, the desired return on investment is often expressed as adesire to receive the same payout regardless of the state that occursamong the set of defined states that comprise the digital option. Forinstance, in the illustrative example of the MSFT stock prices shown inFIG. 6.2.1, a digital call option with strike price of 40 would be, in apreferred embodiment, allocated the same payout irrespective of whichstate of the five states above 40 occurs.

In preferred embodiments of the DBAR DOE of the present invention,traders who invest in digital call options (or strips or spreads)specify a total amount of investment to be made (if the amount is for abuy order) or notional payout to be “sold” (if the amount is for a“sell” order). In a preferred embodiment, the total investment is thenallocated using the compEq( ) multistate allocation method depicted inFIG. 13. In another preferred embodiment, the total amount of the payoutto be received, should the digital option expire in the money, isspecified by the investor, and in a preferred embodiment the investmentamount required to produce such payouts are computed by the multistateallocation method depicted in FIG. 14.

In either embodiment, the investor specifies a desired return oninvestment from a designated set of defined states. A return oninvestment is the amount of value units received from the investmentless the amount of value units invested, divided by the amount invested.In the embodiment depicted in FIG. 13, the amount of value unitsinvested is specified and the amount of value units received, or thepayout from the investment, is unknown until the termination criteriaare fulfilled and the payouts are calculated. In the embodiment depictedin FIG. 14, the amount of value units to be paid out is specified butthe investment amount to achieve that payout it is unknown until thetermination criteria are fulfilled. The embodiment depicted in FIG. 13is known, for example, as a composite trade, and the embodiment depictedin FIG. 14 is known, for example, as a profile trade.

Referring back to FIG. 13, step 1301 invokes a function call to thefunction profEq( ). This function handles those types of trades in whicha desired return-on-investment for a designated set of defined states isspecified by the trader indicating the payout amount to be receivedshould any of the designated set of defined states occur. For example, atrader may indicate that a payout of $10,000 should be received shouldthe MSFT digital calls struck at 40 finish in the money. Thus, if MSFTstock is observed at the expiration date to have a price of 45, theinvestor receives $10,000. If the stock price were to be below 40, theinvestor would lose the amount invested, which is calculated using thefunction profEq( ). This type of desired return-on-investment trade isreferred to as a multistate profile trade, and FIG. 14 depicts thedetailed logical steps for a preferred embodiment of the profEq( )function. In preferred embodiments of a DBAR DOE of the presentinvention, there need not be any profile trades.

Referring back to FIG. 13, step 1302 initializes control loop countervariables. Step 1303 indicates a control loop that executes for eachorder. Step 1304 initializes the variable “norm” to zero and assigns theorder being processed, order[j], to the order data structure. Step 1305begins a control loop that executes for each of defined states thatcomprise a given order. For example, the MSFT digital call option withstrike of 40 illustrated in Table 6.2.1 spans the five states that rangefrom 40 and higher.

In the preferred embodiment depicted in FIG. 13, step 1306 executeswhile the number of states in the order are being processed to calculateof the variable norm, which is the weighted sum of the total investmentsfor each state of the range of defined states which comprise the order.The weights are contained in order.ratio[i], which is a vector typemember of the order data structure illustrated in FIG. 11 as previouslydescribed. For digital call options, whose payout is the same regardlessof the defined state which occurs over the range of states for which thedigital option is in the money, all of the elements of order.ratio[ ]are equal over the range. For trades involving digital strips, theratios in order.ratio[ ] need not be equal. For example, a trader maydesire a payout which is twice as great should a range of states occurcompared to another range of states. The data member order.ratio[ ]would therefore contain information about this desired payout ratio.

In the preferred embodiment depicted in FIG. 13, after all of the statesin the range of states spanning the order have been processed, thecontrol loop counter variable is re-initialized in step 1307, step 1308begins another control loop the defined states spanning the order. Inpreferred embodiments, step 1309 calculates the amount of the investmentspecified by the order that must be invested in each defined statespanning the range of states for the order. Sub-step 2 of step 1309contains the allocation which is assigned to order.invest[i], for eachof these states. This sub-step allocates the amount to be invested in anin-the-money state in proportion to the existing total investment inthat state divided by the sum of all of the investment in thein-the-money states. Sub-steps 3 and 4 of step 1309 add this allocationto the investment totals for each state (contract.stateTotal[state]) andfor all of the states (contract.totallnvested) after subtracting out theallocation from the previous iteration (temp). In this manner, theallocation steps proceed iteratively until a tolerable level of errorconvergence is achieved.

After all of the states in the order have been allocated in 1309, step1310 of the preferred embodiment depicted in FIG. 13 calculates the“price” or implied probability of the order. The “price” of the order isequal to the vector product of the order ratio (a vector quantitycontained in order.ratio[ ]) and the total invested in each state (avector quantity contained in contract.stateTotal[ ]) divided by thetotal amount invested over all of the defined states (contained incontract.totallnvested), after normalization by the maximum value in thevector order.ratio[ ]. As further depicted in step 1310 the resulting“price” for the digital option, strip, or spread is stored in the pricemember of the order data structure (order.price).

In the preferred embodiment of the method of multistate compositeequilibrium calculation for a DBAR DOE of the present invention. Step1311 moves the order processing step to the next order. After all of theorders have been processed, step 1312 of the preferred embodimentdepicted in FIG. 13 calculates the level of error, which is based uponthe percentage deviations of the payouts resulting from the previousiteration to the payouts expected by the trader. If the error istolerably low (e.g., epsilon=10⁻⁸), the compEq( ) function terminates(step 1314). If the error is not tolerably low, then compEq( ) isiterated again, as shown in step 1313.

FIG. 14 depicts a preferred embodiment of a method of multistate profileequilibrium calculation in a DBAR DOE of the present invention. As shownin FIG. 14, when a new multistate profile trade is added, the functionaddProfile( ) of step 1401 adds information about the trade to the datastructure members of the contract data structure, as described above inFIG. 11. The first step of the profEq( ) function, step 1402, shows thatthe profEq( ) function proceeds iteratively until a tolerable level ofconvergence is achieved, i.e., an error below some error parameterepsilon (e.g., 10⁻⁸). If the error objective has not been met, in apreferred embodiment all of the previous allocations from any priorinvocations of profEq( ) are subtracted from the total investments ineach state and from the total investment for all of the states, asindicated in step 1405. This is done for each of the states, asindicated in control loop 1404 after initialization of the loop counter(step 1403).

In a preferred embodiment, the next step, step 1406, computes theinvestment amount necessary to generate the desired return-on-investmentwith a fixed payout profile. Sub-step 1 of 1406 shows that theinvestment amount required to achieve this payout profile for a state isa positive solution to the quadratic equation CDRF 3 set forth inSection 2.4 above. In the preferred embodiment depicted in FIG. 14, thesolution, contract.poTrade[i], is then added to the total investmentamount in that state as indicated in sub-step 2 of step 1406. The totalinvestment amount for all of the states is also increased bycontract.poTrade[i], and sub-step 4 of 1406 increments the control loopcounter for the number of states. In the preferred embodiment depictedin FIG. 14, the calculation of the quadratic equation of sub-step 3 ofstep 1406 is completed for each of the states, and then repeatediteratively until a tolerable level of error is achieved.

FIG. 15 depicts a preferred embodiment of a method for converting “sell”orders to buy orders in a DBAR DOE of the present invention. The methodis contained in the function convertSales( ) called within the limit andmarket order processing steps previously discussed with reference toFIG. 11.

As discussed above in a preferred embodiment of a DBAR DOE, buy ordersand “sell” order are interpreted somewhat differently. The amount of abuy order (as contained in the data structure member order.orderAmount)is interpreted as the amount of the investment to be allocated over therange of states spanning the contingent claim specified in the order.For example, a buy order for 100,000 value units for an MSFT digitalcall with strike price of 60 (order.ratio[ ]=(0,0,0,0,1,1,1) in the MSFTstock example depicted in Table 6.2.1) will be allocated among thestates comprising the order so that, in the case of a digital option,the same payout is received regardless of which constituent state of therange of states is realized. For a “sell” order the order amount (asalso contained in the member data structure order.orderAmount) isinterpreted to be the amount which the trader making the sale stands tolose if the contingent claim (i.e., digital option, spread, or strip)being “sold” expires in the money (i.e., any of the constituent statescomprising the sale order is realized). Thus, the “sale” order amount isinterpreted as a payout (or “notional” or “notional payout”) less theoption premium “sold,” which is the amount that may be lost should thecontingent claim “sold” expire in the money (assuming, that is, theentire order amount can be executed if the order is a limit order). Abuy order, by contrast, has an order amount which is interpreted as aninvestment amount which will generate a payout whose magnitude is knownonly after the termination of trading and the final equilibrium pricesfinalized, should the option expire in the money. Thus, a buy order hasa trade amount which is interpreted as in investment amount or option“premium” (using the language of the conventional options markets)whereas a DBAR DOE “sell” order has an order amount which is interpretedto be a net payout equal to the gross payout lost, should the optionsold expire in the money, less the premium received from the “sale.”Thus, in a preferred embodiment of a DBAR DOE, buy orders have orderamounts corresponding to premium amounts, while “sell” orders have orderamounts corresponding to net payouts.

One advantage of interpreting the order amount of the buy and “sell”orders differently is to facilitate the subsequent “sale” of a buy orderwhich has been executed (in all or part) in a previous trading period.In the case where a subsequent trading period on the same underlyingevent of economic significance or state of a financial product isavailable, a “sale” may be made of a previously executed buy order froma previous and already terminated and finalized trading period, eventhough the observation period may not be over so that it is not knownwhether the option finished in the money. The previously executed buyorder, from the earlier and finalized trading period, has a known payoutamount, should the option expire in the money. This payout amount isknown since the earlier trading period has ended and the finalequilibrium “prices” have been calculated. Once a subsequent tradingperiod on the same underlying event of economic significance is open fortrading (if such a trading period is made available), a trader who hasexecuted the buy order may then sell it by entering a “sell” order withan order. The amount of the “sell” order can be a function of thefinalized payout amount of the buy order (which is now known withcertainty, should the previously bought contingent claim expire in themoney), and the current market price of the contingent claim being“sold.” Setting this order amount of the “sale” equal to y, the tradermay enter a “sale” such that y is equal to:

y=P*(1−q)

where P is the known payout from the previously finalized buy order froma preceding trading period, and q is the “price” of the contingent claimbeing “sold” during the subsequent trading period. In preferredembodiments, the “seller” of the contingent claim in the second periodmay enter in a “sale” order with order amount equal to y and a limit“price” equal to q. In this manner the trader is assured of “selling”his claim at a “price” no worse than the limit “price” equal to q.

Turning now to the preferred embodiment of a method for converting“sale” orders to buy orders depicted in FIG. 15, in step 1501 a controlloop is initiated of orders (contract.numOrders). Step 1502 querieswhether the order under consideration in the loop is a buy(order.buySell=1) or a “sell” order (order.buySell=−1). If the order isa buy order then no conversion is necessary, and the loop is incrementedto the next order as indicated in step 1507.

If, on the other hand, the order is a “sell” order, then in preferredembodiments of the DBAR DOE of the present invention conversion isnecessary. First, the range of states comprising the contingent claimmust be changed to the complement range of states, since a “sale” of agiven range of states is treated as equivalent to a buy order for thecomplementary range of states. In the preferred embodiment of FIG. 15,step 1503 initiates a control loop to execute for each of the definedstates in the contract (contract.numStates), step 1504 does theswitching of the range of states sold to the complementary states to bebought. This is achieved by overwriting the original range of statescontained in order[j].ratio[ ] to a complement range of states. In thispreferred embodiment, the complement is equal to the maximum entry forany state in the original order[ ].ratio[ ] vector (for each order)minus the entry for each state in order[j].ratio[ ]. For example, if atrader has entered an order to sell 50-strike puts in MSFT exampledepicted in Table 6.2.1, then originally order.ratio[ ] is the vector(1,1,1,0,0,0,0), i.e., 1's are entered which span the states (0,30],(30,40], (40,50] and zeroes are entered elsewhere. To obtain thecomplement states to be bought, the maximum entry in the originalorder.ratio[ ] vector for the order is obtained. For the put option tobe “sold,” the maximum of (1,1,1,0,0,0,0) is clearly 1. Each element ofthe original order.ratio[ ] vector is then subtracted from the maximumto produce the complementary states to be bought. For this example, theresult of this calculation is (0,0,0,1,1,1,1), i.e., a purchase of a50-strike call is complementary to the “sale” of the 50-strike put. Iffor example, the original order was for a strip in which the entries inorder.ratio[ ] are not equal, in a preferred embodiment the samecalculation method would be applied. For example, a trader may desire to“sell” a payout should any of the same three states which span the50-strike put occur, but desires to sell a payout of three times theamount of state (40,50] should state (0,30] occur and sell twice thepayout of (40,50] should state (30,40] occur. In this example, theoriginal order.ratio for the “sale” of a strip is equal to(3,2,1,0,0,0,0). The maximum value for any state of this vector is equalto 3. The complementary buy vector is then equal to each element of theoriginal vector subtracted from the maximum, or (0, 1, 2, 3,3,3,3,).Thus, a “sale” of the strip (3,2,1,0,0,0,0) is revised to a purchase ofa strip with order.ratio[ ] equal to (0, 1, 2, 3,3,3,3).

In the preferred embodiment depicted in FIG. 15, after the loop hasiterated through all of the states (the state counter is incremented instep 1505) the loop terminates. After looping through all of the states,the limit order “price” of the “sale” must be revised so that it may beconverted into a complementary buy. This step is depicted in step 1506,where the revised limit order “price” for the complementary buy is equalto one minus the original limit order “price” for the “sell”. Afterfinishing the switching of each state in order.ratio[ ] and setting thelimit order “price” for each order, the loop which increments over theorders goes to the next order, as indicated in step 1507. The conversionof “sell” orders to buy orders terminates when all orders have beenprocessed as indicated in step 1508.

FIG. 16 depicts a preferred embodiment of a method for adjusting limitorders in the presence of transaction fees in a DBAR DOE of the presentinvention. The function which implements this embodiment isfeeAdjustOrders( ) and is invoked in the method for processing limit andmarket orders depicted and discussed with reference to FIG. 11. Limitorder are adjusted for transaction fees to reflect the preference thatorders (after all “sell” orders have been converted to buy orders)should only be executed when the trader specifies that he is willing topay the equilibrium “price,” inclusive of transaction fees. Theinclusion of fees in the “price” produces the “offer” price. Therefore,in a preferred embodiment, all or part of an order with a limit “price”which is greater than or equal to the “offer” price should be executedin the final equilibrium, and an order with a limit “price” lower thanthe “offer” price of the final equilibrium should not be executed atall. To ensure that this equilibrium condition obtains, in a preferredembodiment the limit order “prices” specified by the traders areadjusted for the transaction fee assessed for each order before they areprocessed by the equilibrium calculation, specifically the “add andprune” cycle discussed in Section 6 above and with reference to FIG. 17below, which involves the recomputation of equilibrium “prices.” Thus,the “add and prune” cycle is performed with the adjusted limit order“prices.”

Referring back to FIG. 16, which discloses the steps of the functionfeeAdjustOrders( ) step 1601 initiates a control loop for each order inthe contract (contract.numOrders). The next step 1602 queries whetherthe order being considered is a market order (order.marketLimit=1) or alimit order (order.marketLimit=0). A market order is unconditional andin a preferred embodiment need not be adjusted for the presence oftransaction fee, i.e., it is executed in full regardless of the “offer”side of the market. Thus, if the order is market order, its “limit”price or implied probability is set equal to one as shown in step 1604(order[j].limitPrice=1). If the order being processed in the controlloop of step 1601 is a limit order, then step 1603 revises the initiallimit order by setting the new limit order “price” equal to the initiallimit order “price” less the transaction fee (order.fee). In a preferredembodiment, this function is called after all “sell” orders have beenconverted to buy orders, so that the adjustment for all orders mayinvolve only making the buy orders less likely to be executed byadjusting their respective limit “prices” down by the amount of the fee.After each adjustment is made, the loop over the orders is incremented,as shown in step 1605. After all of the orders have been processed, thefunction feeAdjustOrders( ) terminates as shown in step 1606.

FIG. 17 discloses a preferred embodiment of a method for filling oraddition and removal of lots in a DBAR DOE of the present invention. Thefunction fillRemoveLots( ) which is invoked in the central “add andprune” cycle of FIG. 11, is depicted in detail in FIG. 17. The functionfillRemoveLots( ) implements the method of binary search to determinethe appropriate number of lots to add (or “fill”) or remove. in thepreferred embodiment depicted in FIG. 17, lots are filled or added whenthe function is called with the first parameter equal to 1 and lots areremoved when the function is called with the first parameter equal tozero. The first step of function fillRemoveLots( ) is indicated in step1701. If lots are to be removed, then the method of binary search willtry to find the minimum number of lots to be removed such that the limit“price” of the order (order.limitPrice) is greater than or equal to therecalculated equilibrium “price” (order.price). Thus, if orders are tobe removed, step 1701 sets the maxPremium variable to the number of lotswhich are currently filled in the order, and sets the minPremiumvariable to zero. In other words, in preferred embodiments in a firstiteration the method of binary search will try to find a new number oflots somewhere on the interval between the currently filled number oflots and zero, so that the number of lots to be filled after the step iscompleted is the same or lower than the number of lots currently filled.If lots are to be filled or added, then the method of binary search setsthe maxPremium variable to the order amount (order.amount) since this isthe maximum amount that can be filled for any given order, and theminimum amount equal to the currently filled amount(minPremium=order.filled). That is, if lots are to be filled or added,the method of binary search will try to find the maximum number of lotsthat can be filled or added so that the new number of filled between thecurrent number of lots filled and the number of lots requested in theorder.

In the preferred embodiment depicted in FIG. 17, step 1702 bisects theintervals for binary search created in step 1701 by setting the variablemidPremium equal to the mid point of the interval created in step 1701.A calculation of equilibrium “prices” or implied probabilities for thegroup of DBAR contingent claims equilibrium calculation will then beattempted with the number of lots for the relevant orders reflected bythis midpoint, which will be greater than the current amount filled iflots are to be added and less than the current amount filled if lots areto be removed.

Step 1703 queries whether any change (to within a tolerance) in themid-point of the interval has occurred between the last and currentiteration of the process. If no change has occurred, a new order amountthat can be filled has been found and is revised in step 1708, and thefunction fillRemoveLots( ) terminates in step 1709. If the is differentfrom the midpoint of last iteration, then the new equilibrium iscalculated with the greater (in the case of addition) or lower (in thecase of removal) number of lots as specified in step 1702 of the binarysearch. In a preferred embodiment the equilibrium “prices” arecalculated with these new fill amounts by the multistate allocationfunction, compEq( ), which is described in detail with reference to FIG.13. After the invocation of the function compEq( ), each order will havea current equilibrium “price” as reflected in the data structure memberorder.price. The limit “price” of the order under consideration(order[j])) is then compared to the new equilibrium “price” of the orderunder consideration (order[j].price), as shown in step 1705. If thelimit “price” is worse, i.e., less than the new equilibrium or market“price,” then the binary search has attempted to add too many lots andtries again with fewer lots. The lesser number of lots with which toattempt the next iteration is obtained by setting the new top end of theinterval being bisected to the number of lots just attempted (whichturned out to be too large). This step is depicted in step 1706 of thepreferred embodiment of FIG. 17. With the interval thus redefined andshifted lower, a new midpoint is obtained in step 1702, and a newiteration is performed. If, in step 1705, the newly calculatedequilibrium “price” is less than or equal to the order's limit price,then the binary search will attempt to add or fill additional lots. Inthe preferred embodiment depicted in FIG. 17, the higher number of lotsto add is obtained in step 1707 by setting the lower end of the searchinterval equal to the number of lots for which an equilibriumcalculation was performed in the previous iteration. A new midpoint ofthe newly shifted higher interval is then obtained in step 1702, so thatthe another iteration of the search may be performed with a highernumber of lots. As previously indicated, once further iterations nolonger change the number of lots that are filled, as indicated in step1703, the number of lots of the current iteration is stored, asindicated in step 1708, and the function fillRemoveLots( ) terminates,as indicated in step 1709.

FIG. 18 depicts a preferred embodiment of a method of calculatingpayouts to traders in a DBAR DOE of the present invention, once therealized state corresponding to the event of economic significance orstate of a selected financial product is known. Step 1801 of FIG. 18shows that the predetermined termination criteria with respect to thesubmission of orders by traders have been fulfilled, for example, thetrading period has ended at a previous time (time=t) and the finalcontingent claim prices have been computed and finalized. Step 1802confirms that the event of economic significance or state of a financialproduct has occurred (at a later time=T, where T≧t) and that therealized state is determined to be equal to state k. Thus, according tostep 1802, state k is the realized state. In the preferred embodimentdepicted in FIG. 18, step 1803 initializes a control loop for each orderin the contract (contract.numOrders). For each order, the payout to thetrader is calculated. In preferred embodiments, the payout is a functionof the amount allocated to the realized state (order.invest[k]), theunit payout of the realized state(contract.totalInvested/contract.stateTotal[k]), and the transaction feeof the order as a percentage of the order price (order.fee/order.price).Other methods of allocating payouts net of transaction fees are possibleand would be apparent to one of ordinary skill in the art.

The foregoing detailed description of the figures, and the figuresthemselves, are designed to provide and explain specific illustrationsand examples of the embodiments of methods and systems of the presentinvention. The purpose is to facilitate increased understanding andappreciation of the present invention. The detailed description andfigures are not meant to limit either the scope of the invention, itsembodiments, or the ways in which it may be implemented or practiced. Tothe contrary, additional embodiments and their equivalents within thescope of this invention will be apparent to those of skill in the artfrom reviewing this specification or practicing this invention.

In the embodiment described in Section 7, the DBAR DOE equilibrium iscomputed through a nonlinear optimization to determine the equilibriumexecuted amount for each order, x_(j), in terms of the notional payoutreceived should any state of the set of constituent states of a DBARdigital option occur (defined by B), such that limit orders can beaccepted and processed which are expressed in terms of each trader'sdesired payout (r_(j)). The descriptions of FIGS. 19 and 20 that followexplain this process in detail. Other aspects of this and otherembodiments of the present invention are depicted in FIGS. 21 to 25,referenced in Sections 3, 8 and 9 of this specification.

Generally speaking, in this embodiment, as described in Section 7, theDBAR DOE equilibrium executed amount for the orders is arrived at by:

-   -   (i) inputting into the system how many orders (n) and how many        states (m) are present in the contract;    -   (ii) for each order j, accepting specifications for order or        trade including: (1) if the order is a buy order or a “sell”        order; (2) requested notional payout size (r_(j)); (3) if the        order is market order or limit order; (4) limit order price        (w_(j)) (or if order is market order, then w_(j)=1); (5) the        payout profile or set of defined states for which desired        digital option is in-the money (row j in matrix B); and (6) the        transaction fee (f_(j)).    -   (iii) loading contract and order data structures;    -   (iv) placing opening orders (initial invested premium for each        state, k_(i));    -   (v) converting “sell” orders to complementary buy orders simply        by identifying the range of complementary states being “sold”        and, for each “sell” order j, adjusting the limit “price”        (w_(j)) to one minus the original limit “price” (1-w_(j));    -   (vi) adjusting the limit “price” to incorporate the transaction        fee to produce an adjusted limit price w_(j) ^(a) for each order        j;    -   (vii) grouping the limit orders by placing all of the limit        orders which span or comprise the same range of defined states        into the same group;    -   (viii) sorting the orders upon the basis of the limit order        “prices” from the best (highest “price” buy) to the worst        (lowest “price” buy);    -   (ix) establishing an initial iteration step size, α_(j)(1), the        current step size, α_(j)(κ), will equal the initial iteration        step size, α_(j)(1), until and unless adjusted in step (xii);    -   (x) calculating the equilibrium to obtain the total investment        amount T and the state probabilities, p's, using Newton-Raphson        solution of Equation 7.4.1(b);    -   (xi) computing equilibrium order prices (π_(j)'s) using the p's        obtained in step (viii);    -   (xii) incrementing the orders (x_(j)) which have adjusted limit        prices (w_(j) ^(a)) greater than or equal to the current        equilibrium price for that order (π_(k)) from step (ix) by the        current step size α_(j)(κ);    -   (xiii) decrementing the orders (x_(j)) which have limit prices        (w_(j)) less than the current equilibrium price for that order        (π_(j)) from step (ix) by the current step size α_(j)(κ);    -   (xiv) repeating steps (ix) to (xii) in subsequent iterations        until the values obtained for the executed order notional        payouts achieve a desired convergence, adjusting the current        step size α_(k)(κ) and/or the iteration process after the        initial iteration to further progress towards the desired        convergence;    -   (xv) achieving a desired convergence (along with a final        equilibrium of the prices p's and the total premium invested T)        of the maximum executed notional payout orders x_(j) when        predetermined convergence criteria are met;    -   (xvi) taking the “prices” resulting from the solution final        equilibrium resulting from step (xiii) and adding any applicable        transaction fee to obtain the offer “price” for each respective        contingent claim ordered and subtracting any applicable        transaction fee to obtain the bid “price” for each respective        contingent claim ordered; and    -   (xvii) upon fulfillment of all of the termination criteria        related to the event of economic significance or state of a        selected financial product, allocating payouts to those orders        which have investments on the realized state, where such payouts        are responsive to the final equilibrium “prices” of the orders'        contingent claims and the transaction fees for such orders.

Referring to FIG. 19, illustrative data structures are depicted whichmay be used to store and manipulate the data relevant to the DBAR DOEembodiment described in Section 7 (as well as other embodiments of thepresent invention): data structures for a “contract” (1901), for a“state” (1902) and for an “order” (1903). Each data structure isdescribed below, however it is understood that depending on the actualimplementation of the stepping iterative algorithm, different datamembers or additional data members may be used to solve the optimizationproblem in 7.7.1.

The data structure for a “contract” or group of DBAR contingent claims,shown in 1901, includes data members which store data which are relevantto the construction of the DBAR DOE contract or group of claims underthe embodiment described in Section 7 (as well under other embodimentsof the present invention). Specifically, the contract data structureincludes the following members (also listing the variables denoted bysuch members as described above, if any, and proposed member names forlater programming the stepping iterative algorithm):

-   -   (i) the number of defined states i (m, contract.numStates);    -   (ii) the total premium invested in the contract (T,        contract.totallnvested);    -   (iii) the number of orders j (n, contract.numOrders);    -   (iv) a list of the orders and each order's data (contract.orders        [ ]); and    -   (v) a list of the states and each state's data (contract.states        [ ]).

The data structure for a “state”, shown in 1902, includes data memberswhich store data which are relevant to the construction of each DBAR DOEstate (or spread or strip) under the embodiment described in Section 7,as well as under other embodiments of the present invention.Specifically, each state data structure includes the following members(also listing the variables denoted by such members as described above,if any, and proposed member names for later programming the steppingiterative algorithm):

-   -   (i) the total premium invested in each state i (T_(i),        state.stateTotal);    -   (ii) the executed notional payout per defined state i (y_(i),        state.poReturn[ ]);    -   (iii) the price/probability for each state i (p_(i),        state.statePrice); and    -   (iv) the initial invested premium for each state i to initialize        the contract (k_(i), state.initialState).

The data structure for an “order”, shown in 1903, includes data memberswhich store data which are relevant to the construction of each DBAR DOEorder under the embodiment described in Section 7, as well as underother embodiments of the present invention. Specifically, each orderdata structure includes the following members (also listing thevariables denoted by such members as described above, if any, andproposed member names for later programming the stepping iterativealgorithm):

-   -   (i) the limit price for each order j (w_(j), order.limitPrice);    -   (ii) the executed notional payout per order j, net of fees,        after all predetermined termination criteria have been met        (x_(j), order.executedPayout);    -   (iii) the equilibrium price/probability for each order j (π_(j),        order.orderPrice);    -   (iv) the payout profile for each order j (row j of B,        order.ratio[ ]), specifically it is a vector which specifies the        type of contingent claim to be traded (order.ratio[ ]). For        example, in an embodiment involving a contract with seven        defined states, an order for a digital call option which would        expire in the money should any of the last four states occur        would be rendered in the data member order. ratio[ ] as        order.ratio[0,0,0,1,1,1,1] where the 1's indicate that the same        payout should be generated by the multistate allocation process        when the digital option is in the money, and the 0's indicate        that the option is out of the money, or expires on one of the        respective out of the money states. As another example, a spread        which is in the money should states either states 1,2, 6, or 7        occur would be rendered as ordersatio[1,1,0,0,0,1,1]. As another        example, a digital option strip, which allows a trader to        specify the relative ratios of the final payouts owing to an        investment in such a contingent claim would be rendered using        the ratios over which the strip is in the money. For example, if        a trader desires a strip which pays out three times much as        state 3 should state 1 occur, and twice as much as state 3 if        state 2 occurs, the strip would be rendered as        order.ratio[3,2,1,0,0,0,0]. In other words, the vector stores        integers which indicate the range of states in which an        investment is to be made in order to generate the payout profile        of the contingent claim desired by the trader placing the order.    -   (v) the transaction fee for each order j (f_(j), order.fee);    -   (vi) the requested notional payout per order j (r_(j),        order.requestedPayout);    -   (vii) whether order j is a limit order whose viability for        execution is conditional upon the final equilibrium “price”        being below the limit price after all predetermined termination        criteria have been met, or whether order j is a market order,        which is unconditional (order.marketLimit=0 for a limit order,        =1 for a market order);    -   (viii) whether order j is a buy order or a “sell” order        (order.buySell=1 for a buy, and =−1 for a “sell”); and    -   (ix) the difference between market price and limit price per        order j (g_(j), order.priceGap).

FIG. 20 depicts a logical diagram of the basic steps for limit andmarket order processing in the embodiment of a DBAR DOE described inSection 7, which can be applied to other embodiments of the presentinvention. Step 2001 of FIG. 20 inputs into the system how many orders(contract.numOrders) and how many states (contract.numStates) arepresent in the contract. Then, in step 2002, the computer system acceptsspecifications from the trader or user for each order, including: (1) iforder is a buy order or a “sell” order (order.buySell); (2) requestednotional payout size (order.requestedPayout); (3) if order is marketorder or limit order (order.marketLimit); (4) limit order price(order.limitPrice) (or if order is market order, thenorder.limitPrice=1); (5) the payout profile or set of defined states forwhich desired digital option is in-the money (order.ratio[ ]); and (6)transaction fee (order.fee).

Step 2003 of FIG. 20 loads the relevant data into the contract, stateand order data structures of FIG. 19. The initial value oforder.executedPayout and state.poReturn are set at zero.

Step 2004 initializes the set of DBAR contingent claims, or the“contract,” by placing initial amounts of value units (i.e., initialliquidity) in each state of the set of defined states. The placement ofinitial liquidity avoids a singularity in any of the defined states(e.g., an invested amount in a given defined state equal to zero) whichmay tend to impede multistate allocation calculations. Theinitialization of step 2004 may be done in a variety of different ways.In this embodiment, a small quantity of value units is placed in each ofthe defined states. For example, a single value unit (“lot”) may beplaced in each defined state where the single value unit is expected tobe small in relation to the total amount of volume to be transacted. Instep 2004 of FIG. 20, the initial value units are represented in thevector init[contract.numStates].

In this embodiment, step 2005 of FIG. 20 invokes the functionadjustLimitPrice( ) which converts the limit order price of the “sell”orders to the limit order price of complementary buy orders, and adjuststhe limit order prices to account for the transaction fee charged forthe order (subtracting the fee from the limit order price for a buyorder and subtracting the fee from the converted limit order price for a“sell” order). After the completion of step 2005, all of the limit orderprices for contingent claims—whether buy or “sell” orders, can beprocessed as buy orders together, and the limit order prices areadjusted with fees for the purpose of the ensuing equilibriumcalculations.

In this embodiment, step 2006 groups these buy orders based upon thedistinct ranges of states spanned by the contingent claims specified inthe orders. The range of states comprising the order are contained inthe data member order.ratio[ ] of the order data structure 1903 depictedin FIG. 19. As with the DBAR DOE embodiment discussed in section 6 andFIG. 12 above and other embodiments of the present invention, eachdistinct order[j].ratio[ ] vector in step 2006 in FIG. 20 is groupedseparately from the others in step 2006. Two order[j].ratio[ ] vectorsare distinct for different orders when their difference yields a vectorthat does not contain zero in every element. For example, for a contractwhich contains seven defined states, a digital put option which spansthat first three states has an order[1].ratio[ ] vector equal to(1,1,1,0,0,0,0). A digital call option which spans the last five stateshas an order[2].ratio[ ] vector equal to (0,0,1,1,1,1,1). Because thedifference of these two vectors is equal to (1,1,0,−1,−1,−1,−1), thesetwo orders should be placed into distinct groups, as indicated in step2006.

In this embodiment, step 2006 aggregates orders into relevant groups forprocessing. For the purposes of processing limit orders: (i) all ordersmay be treated as limit orders since orders without limit “price”conditions, e.g., “market orders,” can be rendered as limit buy orders(including “sale” orders converted to buy orders in step 2005) withlimit “prices” of 1, and (ii) all order sizes are processed by treatingthem as multiple orders of the smallest value unit or “lot.”

The relevant groups of step 2006 of FIG. 20 are termed “composite” sincethey may span, or comprise, more than one of the defined states. Forexample, the MSFT Digital Option contract depicted above in Table 6.2.1has defined states (0,30], (30,40], (40,50], (50,60], (60, 70], (70,80], and (80,00]. The 40 strike call options therefore span the fivestates (40,50], (50,60], (60, 70], (70, 80], and (80,00]. A “sale” of a40 strike put, for example, would be aggregated into the same group forthe purposes of step 2004 of FIG. 20, because the “sell” limit order ofa 40 strike put has been converted at step 2005 into a complementary buyorder of a 40 strike call simply by converting the limit order price forthe put order into the complementary limit order price of the callorder.

Similar to step 1206 of DBAR DOE embodiment described with reference toFIG. 12, at the point of step 2007 of this embodiment shown in FIG. 20,all of the orders may be processed as buy orders (because any “sell”orders have been converted to buy orders in step 2005 of FIG. 20) andall limit “prices” have been adjusted (with the exception of marketorders which, in an embodiment of the DBAR DOE or other embodiments ofthe present invention, have a limit “price” equal to one) to reflecttransaction costs equal to the fee specified for the order's contingentclaim (as contained in the data member order[j].fee).

In this embodiment, step 2007 sorts each group's orders based upon theirfee-adjusted limit “prices,” from best (highest “prices”) to worst(lowest “prices”). The grouped orders follow the same aggregation asillustrated in Diagram 1 shown in FIG. 27, and in Section 6. Step 2008establishes an initial iteration step size, init[order.stepSize], thecurrent step size, order.stepSize, will equal the initial iteration stepsize until and unless adjusted in step 2018.

Initially as part of a first iteration (numlteration=1) (2009 a), andlater as part of subsequent iterations, step 2009 invokes the functionfindTotal( ) which calculates the equilibrium of Equation 7.4.7 toobtain the total investment amount (contract.totallnvested) and thestate probabilities (state.statePrice). Step 2010 invokes the functionfindOrderPrices( ) which computes the equilibrium order prices(order.orderPrice) using the state probabilities (state.statePrice)obtained in step 2009. The equilibrium order price for each order(order.orderPrice) is equal to the payout profile for the order(order.ratio[ ]) multiplied with a vector made up of the probabilitiesfor all states i (state. statePrice[contract.numStates]).

Proceeding with the next step of this embodiment depicted in FIG. 20,step 2011 queries whether there is at least a single order which has alimit “price” which is “better” than the current equilibrium “price” forthe ordered option. In this embodiment, for the first iteration of step2011 for a trading period for a group of DBAR contingent claims, thecurrent equilibrium “prices” reflect the placement of the initialliquidity from step 2004. Step 2012 invokes the incrementing( )function, which increments the executed notional payout(order.executedPayout) with the current step size (order.stepSize) foreach order which has a limit price (order.limitPrice) greater than orequal to the current equilibrium price for that order (order.orderPrice)obtained from step 2010 (however, in this embodiment, such incrementingshould not exceed the order's requested payout r_(j)).

Similarly, step 2013 queries whether there is at least a single orderwhich has a limit “price” which is “worse” than the current equilibrium“price” for the ordered option. Step 2014 invokes the decrementing( )function, which decrements the executed notional payout(order.executedPayout) with the current step size (order.stepSize) foreach order which has a limit price (order.limitPrice) less than thecurrent equilibrium price for that order (order.orderPrice) obtainedfrom step 2010 (but, in this embodiment, such decrementing should notproduce an executed order payout below zero).

This embodiment of the DBAR DOE (described in Section 7) simplifies thecomplex comparison and removes the necessity of the “add” and “prune”method for buy and “sell” orders in the DBAR DOE embodiment described inSection 6. In this embodiment (depicted in FIG. 20), once the limitorder price for “sell” orders has been converted to a complementarylimit order price for a buy order, with both types of orders alreadybeing expressed in terms of payout, the notional payout executed foreither a buy or a “sell” order (order.executedPayout) is simplyincremented by the current step size (order.stepSize) if the limit orderprice (order.limitPrice) is greater than or equal to the currentequilibrium price (order.orderPrice), and decremented by the currentstep size (order.stepSize) if the limit order price (order.limitPrice)is less than the current equilibrium price (order.orderPrice).

In step 2015, the counter for the iteration (numIteration) isincremented by 1. Repeat steps 2009 to 2014 for a second iteration(until numIteration=3). Step 2016 queries whether the quantitiescalculated for the executed notional payouts for the orders(order.executedPayout) are converging, and whether the convergence needsto be accelerated. If the executed notional payouts calculated in 2014are not converging or the convergence needs to be accelerated, step 2017queries if the step size (order.stepSize) needs to be adjusted. If thestep size needs to be adjusted, step 2018 adjusts the step size(order.stepSize). Step 2019 queries if the iteration process needs toaccelerated. Step 2020 initiates a linear program if the iterationprocess needs to be accelerated. Then, the iteration process (steps 2009to 2014) is repeated, again.

However, if after step 2016, the quantities calculated for the executednotional payouts for the orders (order.executedPayout) have converged(according to some possibly predetermined or dynamically determinedconvergence criteria), then the iteration process is complete, and thedesired convergence has been achieved in step 2021, along with a finalequilibrium of the order prices (order.orderPrice) and total premiuminvested in the contract (contract.totalInvested), and determination ofthe maximum executed notional payouts for the orders(order.executedPayout).

In step 2022, the order price, not including transaction fees, iscalculated by adding any applicable transaction fee (order.fee) to theequilibrium order price (order.orderPrice) to produce the equilibriumoffer price, and subtracting any applicable transaction fee (order.fee)to the equilibrium order price (order.orderPrice) to produce theequilibrium bid price.

In step 2023, upon fulfillment of all of the termination criteriarelated to the event of economic significance or state of a selectedfinancial product, allocating payouts to those orders which haveinvestments on the realized state, where such payouts are responsive tothe final equilibrium “prices” of the orders' contingent claims(order.orderPrice) and the transaction fees for such orders (order.fee).

The steps and data structures described above and shown in FIGS. 11 to25 for embodiments of DBAR digital options (discussed, for example, inSections 6 and 7 herein) and an embodiment of a demand-based market orauction for structured financial products (discussed, for example, inSection 9 herein), can be implemented within the computer systemdescribed above in reference to FIGS. 1 to 10, as well as in otherembodiments of the present invention. The computer system can includeone or more parallel processors to run, for example, the linear programfor the optimization solution (Section 7), and/or to run one or morefunctions in the DRF or OPF in parallel with a main processor in theacceptance and processing of any DBAR contingent claims, includingdigital options. For DBAR digital options, in addition to determiningand allocating a payout at the end of the trading period, the trader oruser or investor specifies and inputs a desired payout, a selectedoutcome and a limit order price (if any) into the system during thetrading period, and the system determines the investment amount for theorder at the end of the trading period along with an allocation ofpayouts. In other words, the processor and other components (includingcomputer usable medium having computer readable program code, andcomputer usable information storage medium encoded with acomputer-readable data structure) causes the computer system to acceptinputs of information related to a DBAR digital option or to other DBARcontingent claims, perhaps by way of a propagated signal or from aremote terminal by way of the Internet or a private network withdedicated circuits, including each trader's identity, and the desiredpayout, payout profile, and limit price for each order, then throughoutthe trading period the computer system updates the allocation of payoutsper order and the investment amounts per order, and communicates theseupdated amounts to the trader (and, in the case of other DBAR contingentclaims, inputted information may include the investment amount so thatthe computer system can allocate payouts per defined state). At the endof the trading period, the computer system determines a finalizedinvestment amount per order (for DBAR digital options) and allocation ofpayouts per order if the states selected in the order become the statescorresponding to the observed outcome of the event of economicsignificance. In the above DBAR digital option embodiments, the ordersare executed after the end of the trading period at these finalizedamounts. The determination of the investment amount and payoutallocation can be accomplished using any of the embodiments disclosedherein, alone or in combination with each other.

Additionally, the implementations in a computer system (or with anetwork implementation) and the methods described herein to determinethe investment amount and payout allocation as a function of the desiredpayout, selected outcomes, and limit order prices for each order placedin a DBAR digital options market or auction (or to determine the payoutas a function of the selected outcomes, and investment amounts for eachorder in other embodiments of DBAR contingent claims), can be used by abroker to provide financial advice to his/her customers by helping themdetermine when they should invest in a DBAR digital options market orauction based on the type of return they would like to receive, theoutcomes they would like to select, and the limit order price (if any)that they would like to pay or if they should invest in another DBARcontingent claim market or auction based on the amount they would liketo invest, their selected outcomes and other information as describedherein.

Similarly; the implementations and methods described herein can be usedby an investor as a method of hedging for any of the types of economicevents (including any underlying economic events or measured parametersof an underlying economic event as discussed above, including Section3). Hedging involves determining an investment risk in an existingportfolio (even if it includes only one investment) or determining arisk in an asset portfolio (a risk in a lower farm output due to badweather, for example), and offsetting that risk by taking a position ina DBAR digital option or other DBAR contingent claim that has anopposing risk. On the flip side, if a trader is interested in increasingthe risk in an existing portfolio of investments or assets, the DBARdigital option or other DBAR contingent claim is a good tool forspeculation. Again, the trader determines the investment risk in theirasset or investment portfolio, but then takes a position in DBAR digitaloption or other DBAR contingent claim with a similar risk.

The DBAR digital option described above is one type of instrument fortrading in a demand-based market or auction. The digital option setsforth designations of information which are the parameters of the option(like a coupon rate for a Treasury bill), such as the payout profile(corresponding to the selected outcomes for the option to bein-the-money), the desired payout of the option, and the limit orderprice of the option (if any). Other DBAR contingent claims describedabove are other types of instruments for trading in a demand-basedmarket or auction. They set forth parameters including the investmentamount and the payout profile. All instruments are investment vehiclesproviding investment capital into a demand-based market or auction inthe manner described herein.

11. ADVANTAGES OF PREFERRED EMBODIMENTS

This specification sets forth principles, methods, and systems thatprovide trading and investment in groups of DBAR contingent claims, andthe establishment and operation of markets and exchanges for suchclaims. Advantages of the present invention as it applies to the tradingand investment in derivatives and other contingent claims include:

-   (1) Increased liquidity: Groups of DBAR contingent claims and    exchanges for investing in them according to the present invention    offer increased liquidity for the following reasons:    -   (a) Reduced dynamic hedging by market makers. In preferred        embodiments, an exchange or market maker for contingent claims        does not need to hedge in the market. In such embodiments, all        that is required for a well-functioning contingent claims market        is a set of observable underlying real-world events reflecting        sources of financial or economic risk. For example, the quantity        of any given financial product available at any given price can        be irrelevant in a system of the present invention.    -   (b) Reduced order crossing. Traditional and electronic exchanges        typically employ sophisticated algorithms for market and limit        order book bid/offer crossing. In preferred embodiments of the        present invention, there are no bids and offers to cross. A        trader who desires to “unwind” an investment will instead make a        complementary investment, thereby hedging his exposure.    -   (c) No permanent liquidity charge: In the DBAR market, only the        final returns are used to compute payouts. Liquidity variations        and the vagaries of execution in the traditional markets do not,        in preferred embodiments, impose a permanent tax or toll as they        typically do in traditional markets. In any event, in preferred        embodiments of the present invention, liquidity effects of        amounts invested in groups of DBAR claims are readily calculable        and available to all traders. Such information is not readily        available in traditional markets.-   (2) Reduced credit risk: In preferred embodiments of the present    invention, the exchange or dealer has greatly increased assurance of    recovering its transaction fee. It therefore has reduced exposure to    market risk. In preferred embodiments, the primary function of the    exchange is to redistribute returns to successful investments from    losses incurred by unsuccessful investments. By implication, traders    who use systems of the present invention can enjoy limited    liability, even for short positions, and a diversification of    counterparty credit risk.-   (3) Increased Scalability: The pricing methods in preferred    embodiments of systems and methods of the present invention for    investing in groups of DBAR contingent claims are not tied to the    physical quantity of underlying financial products available for    hedging. In preferred embodiments an exchange therefore can    accommodate a very large community of users at lower marginal costs.-   (4) Improved Information Aggregation: Markets and exchanges    according to the present invention provide mechanisms for efficient    aggregation of information related to investor demand, implied    probabilities of various outcomes, and price.-   (5) Increased Price Transparency: Preferred embodiments of systems    and methods of the present invention for investing in groups of DBAR    contingent claims determine returns as functions of amounts    invested. By contrast, prices in traditional derivatives markets are    customarily available for fixed quantities only and are typically    determined by complex interactions of supply/demand and overall    liquidity conditions. For example, in a preferred embodiment of a    canonical DRF for a group of DBAR contingent claims of the present    invention, returns for a particular defined state are allocated    based on a function of the ratio of the total amount invested across    the distribution of states to the amount on the particular state.-   (6) Reduced settlement or clearing costs: In preferred embodiments    of systems and methods for investing in groups of DBAR contingent    claims, an exchange need not, and typically will not, have a need to    transact in the underlying physical financial products on which a    group of DBAR contingent claims may be based. Securities and    derivatives in those products need not be transferred, pledged, or    otherwise assigned for value by the exchange, so that, in preferred    embodiments, it does not need the infrastructure which is typically    required for these back office activities.-   (7) Reduced hedging costs: In traditional derivatives markets,    market makers continually adjust their portfolio of risk exposures    in order to mitigate risks of bankruptcy and to maximize expected    profit. Portfolio adjustments, or dynamic hedges, however, are    usually very costly. In preferred embodiments of systems and methods    for investing in groups of DBAR contingent claims, unsuccessful    investments hedge the successful investments. As a consequence, in    such preferred embodiments, the need for an exchange or market maker    to hedge is greatly reduced, if not eliminated.-   (8) Reduced model risk: In traditional markets, derivatives dealers    often add “model insurance” to the prices they quote to customers to    protect against unhedgable deviations from prices otherwise    indicated by valuation models. In the present invention, the price    of an investment in a defined state derives directly from the    expectations of other traders as to the expected distribution of    market returns. As a result, in such embodiments, sophisticated    derivative valuation models are not essential. Transaction costs are    thereby lowered due to the increased price transparency and    tractability offered by the systems and methods of the present    invention.-   (9) Reduced event risk: In preferred embodiments of systems and    methods of the present invention for investing in groups of DBAR    contingent claims, trader expectations are solicited over an entire    distribution of future event outcomes. In such embodiments,    expectations of market crashes, for example, are directly observable    from indicated returns, which transparently reveal trader    expectations for an entire distributions of future event outcomes.    Additionally, in such embodiments, a market maker or exchange bears    greatly reduced market crash or “gap” risk, and the costs of    derivatives need not reflect an insurance premium for discontinuous    market events.-   (10) Generation of Valuable Data: Traditional financial product    exchanges usually attach a proprietary interest in the real-time and    historical data that is generated as a by-product from trading    activity and market making. These data include, for example, price    and volume quotations at the bid and offer side of the market. In    traditional markets, price is a “sufficient statistic” for market    participants and this is the information that is most desired by    data subscribers. In preferred embodiments of systems and methods of    the present invention for investing in groups of DBAR contingent    claims, the scope of data generation may be greatly expanded to    include investor expectations of the entire distribution of possible    outcomes for respective future events on which a group of DBAR    contingent claims can be based. This type of information (e.g., did    the distribution at time t reflect traders' expectations of a market    crash which occurred at time t+1?) can be used to improve market    operation. Currently, this type of distributional information can be    derived only with great difficulty by collecting panels of option    price data at different strike prices for a given financial product,    using the methods originated in 1978 by the economists Litzenberger    and Breeden and other similar methods known to someone of skill in    the art. Investors and others must then perform difficult    calculations on these data to extract underlying distributions. In    preferred embodiments of the present invention, such distributions    are directly available.-   (11) Expanded Market for Contingent Claims: Another advantage of the    present invention is that it enables a well functioning market for    contingent claims. Such a market enables traders to hedge directly    against events that are not readily hedgable or insurable in    traditional markets, such as changes in mortgage payment indices,    changes in real estate valuation indices, and corporate earnings    announcements. A contingent claims market operating according to the    systems and methods of the present invention can in principle cover    all events of economic significance for which there exists a demand    for insurance or hedging.-   (12) Price Discovery: Another advantage of systems and methods of    the present invention for investing in groups of DBAR contingent    claims is the provision, in preferred embodiments, of a returns    adjustment mechanism (“price discovery”). In traditional capital    markets, a trader who takes a large position in relation to overall    liquidity often creates the risk to the market that price discovery    will break down in the event of a shock or liquidity crisis. For    example, during the fall of 1998, Long Term Capital Management    (LTCM) was unable to liquidate its inordinately large positions in    response to an external shock to the credit market, i.e., the    pending default of Russia on some of its debt obligations. This risk    to the system was externalized to not only the creditors of LTCM,    but also to others in the credit markets for whom liquid markets    disappeared. By contrast, in a preferred embodiment of a group of    DBAR contingent claims according to the present invention, LTCM's    own trades in a group of DBAR contingent claims would have lowered    the returns to the states invested in dramatically, thereby reducing    the incentive to make further large, and possibly destabilizing,    investments in those same states. Furthermore, an exchange for a    group of DBAR contingent claims according to the present invention    could still have operated, albeit at frequently adjusted returns,    even during, for example, the most acute phases of the 1998 Russian    bond crisis. For example, had a market in a DBAR range derivative    existed which elicited trader expectations on the distribution of    spreads between high-grade United States Treasury securities and    lower-grade debt instruments, LTCM could have “hedged” its own    speculative positions in the lower-grade instruments by making    investment in the DBAR range derivatives in which it would profit as    credit spreads widened. Of course, its positions by necessity would    have been sizable thereby driving the returns on its position    dramatically lower (i.e., effectively liquidating its existing    position at less favorable prices). Nevertheless, an exchange    according to preferred embodiments of the present invention could    have provided increased liquidity compared to that of the    traditional markets.-   (13) Improved Offers of Liquidity to the Market: As explained above,    in preferred embodiments of groups of DBAR contingent claims    according to the present invention, once an investment has been made    it can be offset by making an investment in proportion to the    prevailing traded amounts invested in the complement states and the    original invested state. By not allowing trades to be removed or    cancelled outright, preferred embodiments promote two advantages:    -   (1) reducing strategic behavior (“returns-jiggling”)    -   (2) increasing liquidity to the market    -   In other words, preferred embodiments of the present invention        reduce the ability of traders to make and withdraw large        investments merely to create false-signals to other participants        in the hopes of creating last-minute changes in closing returns.        Moreover, in preferred embodiments, the liquidity of the market        over the entire distribution of states is information readily        available to traders and such liquidity, in preferred        embodiments, may not be withdrawn during the trading periods.        Such preferred embodiments of the present invention thus provide        essentially binding commitments of liquidity to the market        guaranteed not to disappear.-   (14) Increased Liquidity Incentives: In preferred embodiments of the    systems and methods of the present invention for trading or    investing in groups of DBAR contingent claims, incentives are    created to provide liquidity over the distribution of states where    it is needed most. On average, in preferred embodiments, the implied    probabilities resulting from invested amounts in each defined state    should be related to the actual probabilities of the states, so    liquidity should be provided in proportion to the actual    probabilities of each state across the distribution. The traditional    markets do not have such ready self-equilibrating liquidity    mechanisms—e.g., far out-of-the-money options might have no    liquidity or might be excessively traded. In any event, traditional    markets do not generally provide the strong (analytical)    relationship between liquidity, prices, and probabilities so readily    available in trading in groups of DBAR contingent claims according    to the present invention.-   (15) Improved Self-Consistency: Traditional markets customarily have    “no-arbitrage” relationships such as put-call parity for options and    interest-rate parity for interest rates and currencies. These    relationships typically must (and do) hold to prevent risk-less    arbitrage and to provide consistency checks or benchmarks for    no-arbitrage pricing. In preferred embodiments of systems and    methods of the present invention for trading or investing in groups    of DBAR contingent claims, in addition to such “no-arbitrage”    relationships, the sum of the implied probabilities over the    distribution of defined states is known to all traders to equal    unity. Using the notation developed above, the following relations    hold for a group of DBAR contingent claims using a canonical DRF:

$\begin{matrix}{r_{i} = {\frac{\left( {1 - f} \right)*{\sum\limits_{i}T_{i}}}{T_{i}} - 1}} & (a) \\{q_{i} = {\frac{1 - f}{r_{i} + 1} = \frac{T_{i}}{\sum\limits_{i}T_{i}}}} & (b) \\{{\sum\limits_{i}q_{i}} = 1} & (c)\end{matrix}$

-   -   In other words, in a preferred embodiment, the sum across a        simple function of all implied probabilities is equal to the sum        of the amount traded for each defined state divided by the total        amount traded. In such an embodiment, this sum equals 1. This        internal consistency check has no salient equivalent in the        traditional markets where complex calculations are typically        required to be performed on illiquid options price data in order        to recover the implied probability distributions.

-   (16) Facilitated Marginal Returns Calculations: In preferred    embodiments of systems and methods of the present invention for    trading and investing in groups of DBAR contingent claims, marginal    returns may also be calculated readily. Marginal returns r^(m) are    those that prevail in any sub-period of a trading period, and can be    calculated as follows:

$\begin{matrix}{r_{i,{t - 1},t}^{m} = \frac{{r_{i,t}*T_{i,t}} - {r_{i,{t - 1}}*T_{i,{t - 1}}}}{T_{i,t} - T_{i,{t - 1}}}} & (1)\end{matrix}$

-   -   where the left hand side is the marginal returns for state i        between times t−1 and t; r_(i,t) and r_(i,t−1) are the unit        returns for state i at times t, and t−1, and T_(i,t) and        T_(i,t−1) are the amounts invested in state i at times t and        t−1, respectively.

In preferred embodiments, the marginal returns can be displayed toprovide important information to traders and others as to the adjustmentthroughout a trading period. In systems and methods of the presentinvention, marginal returns may be more variable (depending on the sizeof the time increment among other factors) than the returns which applyto the entire trading period.

-   (17) Reduced Influence By Market Makers: In preferred embodiments of    the systems and methods of the present invention, because returns    are driven by demand, the role of the supply side market maker is    reduced if not eliminated. A typical market maker in the traditional    markets (such as an NYSE specialist or a swaps book-runner)    typically has privileged access to information (e.g., the limit    order book) and potential conflicts of interest deriving from dual    roles as principal (i.e., proprietary trader) and market maker. In    preferred embodiments of the present invention, all traders have    greater information (e.g., investment amounts across entire    distribution of states) and there is no supply-side conflict of    interest.-   (18) Increased Ability to Generate and Replicate Arbitrary Payout    Distributions: In preferred embodiments of the systems and methods    of the present invention for investing and trading in groups of DBAR    contingent claims, traders may generate their own desired    distributions of payouts, i.e., payouts can be customized very    readily by varying amounts invested across the distribution of    defined states. This is significant since groups of DBAR contingent    claims can be used to readily replicate payout distributions with    which traders are familiar from the traditional markets, such as    long stock positions, long and short futures positions, long options    straddle positions, etc. Importantly, as discussed above, in    preferred embodiments of the present invention, the payout    distributions corresponding to such positions can be effectively    replicated with minimal expense and difficulty by having a DBAR    contingent claim exchange perform multi-state allocations. For    example, as discussed in detail in Section 6 and with reference to    FIGS. 11-18, in preferred embodiments of the system and methods of    the present invention, payout distributions of investments in DBAR    contingent claims can be comparable to the payout distributions    expected by traders for purchases and sales of digital put and call    options in conventional derivatives markets. While the payout    distributions may be comparable, the systems and methods of the    present invention, unlike conventional markets, do not require the    presence of sellers of the options or the matching of buy and sell    orders.-   (19) Rapid implementation: In various embodiments of the systems and    methods of the present invention for investing and trading in groups    of DBAR contingent claims, the new derivatives and risk management    products are processed identically to derivative instruments traded    in the over-the-counter (OTC) markets, regulated identically to    derivative instruments traded in the OTC markets and conform to    credit and compliance standards employed in OTC derivatives markets.    The product integrates with the practices, culture and operations of    existing capital and asset markets as well as lends itself to    customized applications and objectives.

In addition to the above advantages, the demand-based trading system mayalso provide the following benefits:

-   -   (1) Aggregation of liquidity: Fragmentation of liquidity, which        occurs when trading is spread across numerous strike prices, can        inhibit the development of an efficient options market. In a        demand-based market or auction, no fragmentation occurs because        all strikes fund each other. Interest in any strike provides        liquidity for all other strikes. Batching orders across time and        strike price into a demand-based limit order book is an        important feature of demand-based trading technology and is the        primary means of fostering additional liquidity.    -   (2) Limited liability: A unique feature of demand-based trading        products is their limited liability nature. Conventional options        offer limited liability for purchases only. Demand-based trading        digital options and other DBAR contingent claims have the        additional benefit of providing a known, finite liability to        option sellers, based on the notional amount of the option        traded. This will provide additional comfort for short sellers        and consequently will attract additional liquidity, especially        for out-of-the money options.    -   (3) Visibility/Transparency: Customers trading in demand-based        trading products can gain access to unprecedented transparency        when entering and viewing orders. Prices for demand-based        trading products (such as digital options) at each strike price        can be displayed at all times, along with the volume of orders        that would be cleared at the indicated price. A limit order book        displaying limit orders by strike can be accessible to all        customers. Finally, the probability distribution resulting from        all successful orders in the market or auction can be displayed        in a familiar histogram form, allowing market participants to        see the market's true consensus estimate for possible future        outcomes.        -   Demand-based trading solutions can use digital options,            which may have advantages for measuring market expectations:            the price of the digital option is simply the consensus            probability of the specific event occurring. Since the            interpretation of the pricing is direct, no model is            required and no ambiguity exists when determining market            expectations.    -   (4) Efficiency: Bid/Ask spreads in demand-based trading products        can be a fraction of those for options in traditional markets.        The cost-efficient nature of the demand-based trading mechanism        translates directly into increased liquidity available for        taking positions.    -   (5) Enhancing returns with superior forecasting: Managers with        superior expertise can benefit from insights, generating        significant incremental returns without exposure to market        volatility. Managers may find access to digital options and        other DBAR contingent claims useful for dampening the effect of        short-term volatility of their underlying portfolios.    -   (6) Arbitrage: Many capital market participants engage in        macroeconomic ‘arbitrage.’ Investors with skill in economic and        financial analysis can detect imbalances in different sectors of        the economy, or between the financial and real economies, and        exploit them using DBAR contingent claims, including, for        example, digital options, based on economic events, such as        changes in values of economic statistics.

12. TECHNICAL APPENDIX

This technical appendix provides the mathematical foundation underlyingthe computer code listing of Table 1: Illustrative Visual Basic ComputerCode for Solving CDRF 2. That computer code listing implements aprocedure for solving the Canonical Demand Reallocation Function (CDRF2) by preferred means which one of ordinary skill in the art willrecognize are based upon the application of a mathematical method knownas fixed point iteration.

As previously indicated in the specification, the simultaneous systemembodied by CDRF 2 does not provide an explicit solution and typicallywould require the use of numerical methods to solve the simultaneousquadratic equations included in the system. In general, such systemswould typically be solved by what are commonly known as “grid search”routines such as the Newton-Raphson method, in which an initial solutionor guess at a solution is improved by extracting information from thenumerical derivatives of the functions embodied in the simultaneoussystem.

One of the important advantages of the demand-based trading methods ofthe present invention is the careful construction of CDRF 2 which allowsfor the application of fixed point iteration as a means for providing anumerical solution of CDRF 2. Fixed point iteration means are generallymore reliable and computationally less burdensome than grid searchroutines, as the computer code listing in Graph 12.1 shown in FIG. 28illustrates.

Fixed Point Iteration

The solution to CDRF 2 requires finding a fixed point to a system ofequations. Fixed points represent solutions since they convey theconcept of a system at “rest” or equilibrium, i.e., a fixed point of asystem of functions or transformations denoted g(a) exists if

a=g(a)

Mathematically, the function g(a) can be said to be a map on the realline over the domain of a. The map, g(x), generates a new point, say, y,on the real line. If x=y, then x is called a fixed point of the functiong(a). In terms of numerical solution techniques, if g(a) is a non-linearsystem of equations and if x is a fixed point of g(a), then a is alsothe zero of the function. If no fixed points such as x exist for thefunction g(a), then grid search type routines can be used to solve thesystem (e.g., the Newton-Raphson Method, the Secant Method, etc.). If afixed point exists, however, its existence can be exploited in solvingfor the zero of a simultaneous non-linear system, as follows.

Choose an initial starting point, x₀, which is believed to be somewherein the neighborhood of the fixed point of the function g(a). Then,assuming there does exist a fixed point of the function g(a), employ thefollowing simple iterative scheme:

x _(i+1) =g(x _(i)), where x ₀ is chosen as starting point

where i=0, 1, 2, . . . n. The iteration can be continued until a desiredprecision level, ε, is achieved, i.e.,

x _(n) =g(x _(n−1)), until |g(x _(n−1))−x _(n)<ε

The question whether fixed point iteration will converge, of course,depends crucially on the value of the first derivative of the functiong(x) in the neighborhood of the fixed point as shown in FIG. 28. Aspreviously indicated, an advantage of the present invention is theconstruction of CDRF 2 in such a way so that it may be represented interms of a multivariate function, g(x), which is continuous and has aderivative whose value is between 0 and 1, as shown below.

Fixed Point Iteration as Applied to CDRF 2

This section will demonstrate that (1) the system of equations embodiedin CDRF 2 possesses a fixed point solution and (2) that this fixed pointsolution can be located using the method of fixed point iterationdescribed in Section A, above.The well known fixed point theorem provides that, if g: [a, b]—>[a, b]is continuous on [a, b] and differentiable on (a, b) and there is aconstant k<1 such that for all x in (a, b),

|g′(x)|≦k

then g has a unique fixed point x* in [a, b]. Moreover, for any x in [a,b] the sequence defined by

x ₀ =x and x _(n+1) =g(x _(n))

converges to x* and for all n

${{x_{n} - x^{*}}} \leq {\frac{k^{n}*{{x_{1} - x_{0}}}}{1 - k}.}$

The theorem can be applied CDRF 2 as follows. First, CDRF 2 in apreferred embodiment relates the amount or amounts to be invested acrossthe distribution of states for the CDRF, given a payout distribution, byinverting the expression for the CDRF and solving for the traded amountmatrix A:

A=P*Π(A,f)⁻¹  (CDRF 2)

CDRF 2 may be rewritten, therefore, in the following form:

A=g(A)

where g is a continuous and differentiable function. By theaforementioned fixed point theorem, CDRF 2 may be solved by means offixed point iteration if:

g(A)<1

i.e., the multivariate function g(A) has a first derivative less than 1.Whether g(A) has a derivative less than 1 with respect to A can beanalyzed as follows. As previously indicated in the specification, forany given trader and any given state I, CDRF2 contains equations of thefollowing form relating the desired payout p (assumed to be greater than0) to the traded amount a required to generate the desired payout, givena total traded amount already traded for state i of T_(i) (also assumedto be greater than 0) and the total traded amount for all the states ofT:

$\alpha = {\left( \frac{T_{i} + \alpha}{T + \alpha} \right)*p}$so that

${g(\alpha)} = {\left( \frac{T_{i} + \alpha}{T + \alpha} \right)*p}$

Differentiating g(α) with respect to α yields:

${g^{\prime}(\alpha)} = {\left( \frac{T - T_{i}}{T + \alpha} \right)*\frac{p}{T + \alpha}}$

Since the DRF Constraint defined previously in the specificationrequires that payout amount p not exceed the total amount traded for allof the states, the following condition holds:

$\frac{p}{T + \alpha} \leq 1$and therefore since

$\left( \frac{T - T_{i}}{T + \alpha} \right) < 1$it is the case that

0<g′(α)<1

so that the solution to CDRF 2 can be obtained by means of fixed pointiteration as embodied in the computer code listing of Table 1.

13. CONCLUSION

Preferred embodiments of the invention have been described in detailabove, various changes thereto and equivalents thereof will be readilyapparent to one of ordinary skill in the art and are encompassed withinthe scope of this invention and the appended claim. For example, manytypes of demand reallocation functions (DRFs) can be employed to financegains to successful investments with losses from unsuccessfulinvestments, thereby achieving different risk and return profiles totraders. Additionally, this disclosure has primarily discussed methodsand systems for groups and portfolios of DBAR contingent claims, andmarkets and exchanges and auctions for those groups. The methods andsystems of the present invention can readily be adapted by financialintermediaries for use within the traditional capital and insurancemarkets. For example, a group of DBAR contingent claims can be embeddedwithin a traditional security, such as a bond for a given corporateissuer, and underwritten and issued by an underwriter as previouslydiscussed. It is also intended that such embodiments and theirequivalents are encompassed by the present invention and the appendedclaims.

The present invention has been described above in the context of tradingderivative securities, specifically the implementation of an electronicderivatives exchange which facilitates the efficient trading of (i)financial-related contingent claims such as stocks, bonds, andderivatives thereon, (ii) non-financial related contingent claims suchas energy, commodity, and weather derivatives, and (iii) traditionalinsurance and reinsurance contracts such as market loss warranties forproperty-casualty catastrophe risk. The present invention has also beendescribed above in the context of a DBAR digital options exchange, andin the context of offering DBAR-enabled financial products. The presentinvention is not limited to these contexts, however, and can be readilyadapted to any contingent claim relating to events which are currentlyuninsurable or unhedgable, such as corporate earnings announcements,future semiconductor demand, and changes in technology.

In the preceding specification, the present invention has been describedwith reference to specific exemplary embodiments thereof. It will,however, be evident that various modifications and changes may be madethereunto without departing from the broader spirit and scope of thepresent invention as set forth in the claims that follow. Thespecification and drawings are accordingly to be regarded in anillustrative rather than restrictive sense.

1-245. (canceled)
 246. A method of an investment management machine forassisting in conducting demand-based trading, comprising: receiving, bythe investment management machine and via a computer network, anindication of at least one parameter of a financial product; anddetermining and outputting, by a processor of the investment managementmachine, and substantially in real time, a selected outcome, a desiredpayout, and an investment amount of each contingent claim in a set of atleast one contingent claim as a function of the at least one parameterof the financial product; wherein the investment amount is pledged to bepaid, such pledge occurring prior to any of the selected outcomes of thecontingent claim, and the desired payouts are payable conditional upon aprior occurrence of the respective selected outcome.
 247. A method of aninvestment management machine for assisting in investing, comprising:obtaining, by an investment management machine, user-input representinga selected outcome and a desired payout, the selected outcomecorresponding to at least one of a plurality of states, each statecorresponding to at least one possible outcome of an event of economicsignificance; and determining and outputting, by a processor of theinvestment management machine, an investment amount as a function of theselected outcome, the desired payout and a total amount invested in theplurality of states, and outputting the investment amount; wherein theinvestment amount is pledged to be paid, such pledge occurring prior tothe event of economic significance, and the desired payout is payableconditional upon and after occurrence of the selected outcome.
 248. Amethod of an investment management machine for assisting in conductingdemand-based trading, comprising: dividing a range of possible outcomesof an event of economic significance into a plurality of states, eachstate representing a respective subset of the range of possibleoutcomes, wherein a respective subset of at least one of the statesincludes more that one possible outcome; storing, by a processor of theinvestment management machine and in a storage device of the investmentmanagement machine, the plurality of states; receiving, by theinvestment management machine and via a computer network, a plurality ofinvestment orders, each of at least a subset of the plurality ofinvestment orders indicating a respective desired payout and arespective selected subset of the range of possible outcomes,corresponding to at least one respective one of the plurality of states,in which to invest, wherein the indicated respective selected subsets ofthe range of possible outcomes differ between at least some of theplurality of investment orders; for at least one of the orders,determining, by the processor and substantially in real time, aninvestment amount for the order as a function of (a) the selected subsetof the range of possible outcomes and the desired payout indicated bythe respective order and (b) a total amount invested in the plurality ofstates; and outputting the investment amount; wherein the investmentamount is pledged to be paid, such pledge occurring prior to the eventof economic significance, and, for each of the at least a subset of theplurality of investment orders, the respective desired payout is payableconditional upon a prior occurrence of the respective selected subset ofthe range of possible outcomes.
 249. A method of an investmentmanagement machine for assisting in conducting demand-based trading,comprising: receiving, by the investment management machine and via acomputer network, an indication of at least one parameter of a financialproduct; determining, by a processor of the investment managementmachine, for one of a plurality of investors, and substantially in realtime, a selected outcome, a desired payout, and an investment amount ofeach contingent claim in a set of at least one contingent claim as afunction of the at least one parameter of the financial product; andoutputting the selected outcomes, the desired payouts, and theinvestment amounts; wherein the investment amounts are pledged to bepaid, such pledges occurring prior to the outcome of the contingentclaim, and, for each of the at least one contingent claim, therespective desired payout is payable conditional upon and afteroccurrence of the respective selected outcome.